Investigation of line shapes and line intensities by high-resolution UV-photoemission spectroscopy — Some case studies on noble-metal surfaces

Investigation of line shapes and line intensities by high-resolution UV-photoemission spectroscopy Some case studies on noble-metal surfaces

Rend Matzdorf Fachbereich Physik der Universit~it Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany

ELSEVIER

Amsterdam-Lausanne-New York-Oxford-Shannon-Tokyo

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R. Matzdorf /Surface Science Reports 30 (1998) 153-206

Contents 1. Introduction 2. Experimental requirements for highest resolution 2.1. Energy resolution function of a hemispherical analyzer 2.2. Electrostatic lens systems 2.3. Light sources 2.4. Design of an optimized system 3. Line shape of bulk-emission peaks 3.1. The photoemission process 3.2. Lifetimes of photoelectron and photohole 3.3. Experiments on a low index surface 3.4. Experiments on a vicinal surface 3.5. Final-state band-structure effects 3.6. One-step calculations in the Green's function formalism 3.7. Electron-phonon scattering 3.8. Matrix elements 3.9. Surface emission 4. Line shape of surface states 4.1. Historical outline 4.2. Relevance for lifetime investigations 4.3. Modification of line shapes by instrumental resolution 4.4. Electron-phonon interaction 4.5. Lifetime of the hole state 4.6. Defect and impurity scattering 4.7. Lifetime of image potential states 5. Summary and conclusions

155 157 157 159 161 163 164 164 165 167 171 174 177 182 184 185 186 186 189 189 191 193 197 199 200

Acknowledgements References

201 201

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surface science reports ELSEVIER

Surface Science Reports 30 (1998) 153-206

Investigation of line shapes and line intensities by high-resolution UV-photoemission spectroscopy - Some case studies on noble-metal surfaces Rend Matzdorf Fachbereich Physik der Universitiit Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany

Manuscript received in final form 13 August 1997

Abstract Line shapes in angle-resolved photoemission spectra from solid surfaces contain a wealth of information about manybody effects in the electron system. However, an analysis of experimental data depends on a very detailed understanding of the contributions to line width due to various other effects. This report reviews photoemission line shape studies of bulk- and surface-emission peaks measured on noble-metal surfaces and their interpretation in terms of electron and photohole lifetimes, electron-phonon interaction and defect scattering. Special attention is focussed on the possibility to measure the photohole lifetime in the vicinity of the Fermi level. Different channels of photohole decay like Auger processes, phonon creation and elastic scattering at surface defects and impurities are discussed along with highresolution spectra from surface states. We discuss to which extent the various effects can be distinguished experimentally. Information can also be extracted from experimental data concerning properties of the many-electron system at elevated temperature.

1.

Introduction

In the last two decades angle-resolved photoelectron spectroscopy has developed into one of the most important tools for studying the electronic structure of solids [1-5]. Using photon energies between 5 and about 50 eV the three-dimensional valence band structure of the bulk as well as the two-dimensional dispersion of surface state bands can be measured directly. In typical experiments energy distribution curves are measured as a function of photon energy and electron emission angle. In a kinematic analysis exploiting energy and momentum conservation laws the energy of wave vector relation E(k) is determined. In these band-structure measurements energetic peak positions are analyzed whereas the shape and the intensity of the photoemission peaks are generally not interpreted at all. In some experiments " o n - o f f " variations in peak intensity as a function of the light polarization have been 0167-5729/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII SO167-5729(97)00013-7

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R. Matzdorf/Surface Science Reports 30 (1998) 153-206

analyzed to investigate the point-group symmetries of the electron bands [ 1-5]. However, there are only few examples in the literature where matrix elements have been measured and compared to calculations. This is due to the additional, often strong dependence of peak intensities on sample temperature and surface quality, which is not well understood and was studied only for few cases in recent years. Meanwhile, photoelectron spectroscopy has progressed [6] to a point where measured line width and line shape are no longer determined exclusively by instrumental resolution but also by intrinsic mechanisms such as the lifetimes of the created photohole and photoelectron. Since many years [7] special interest is focussed on the exploration of this additional spectroscopic information with the intention to measure the inverse photohole lifetime Fh. This quantity is generally not accessible by other techniques and no comprehensive picture of it has been obtained yet. Fh provides a unique probe of many-body processes operating within the band and reflects the degree to which the band structure itself is well defined. Density-of-state arguments force the lifetime of the hole to vary quadratically with energy at about the Fermi level in a metal [7]. There were several attempts in the past to measure this functional form of the quasiparticle lifetime in the vicinity to the Fermi level. In this field in particular surface states were used as probes for several reasons. But there are severe pitfalls in the interpretation of angle-resolved photoemission data and Smith [8] stated recently that experimental photoemission techniques are not yet sufficiently developed to measure the hole lifetime in the energy range around the Fermi energy. In the context of the investigation of high-To superconductors, and of solids with an electronic structure of dimensions below 3, it is crucial to understand to which extent angle-resolved photoemission is able to measure a true spectral function of quasiparticles close to the Fermi level [8]. For this discussion of the capability of photoelectron spectroscopy to measure lifetimes and to investigate other correlation effects in general, detailed studies of all effects influencing the photoemission peak shapes are therefore of considerable interest. In this report the influence on peak shapes and intensities of experimental energy and angular resolution, initial- and final-state lifetime as well as phonon and defect scattering are discussed. Various experimental results taken from noble-metal surfaces are reported. These metals are ideal as test substances since their bulk and surface band structures are well-known and the preparation of clean and well-ordered surfaces in ultra-high vacuum is comparatively easy. The experimental results are compared to one-step photoemission calculations as well as to calculations modeling the effects due to final-state lifetime, phonon scattering, instrumental resolution, and surface state band modifications due to defect scattering. This report is organized as follows. In Section 2 the experimental equipment required for highquality experiments in particular with respect to the energy and angular resolution functions is discussed. In Section 3 the photoemission process is discussed generally with respect to lifetime effects and to experimental resolution. By incorporation of these two broadening mechanisms the shape of bulk-emission spectra is modeled. Final-state scattering effects due to phonons and surface step edges are investigated and intensities as determined from experiment are compared to one-step calculations. In Section 4 attention is focussed on the surface states. Their photoemission peak shapes are not influenced by the final-state lifetime which may be a great advantage in studies of hole lifetimes. In this context the requirements in instrumental resolution and surface quality are discussed. Contributions to the quasiparticle lifetime due to Auger decay, electron-phonon interaction and elastic scattering at defects will be discussed in some detail. Finally Section 5 summarizes the most important conclusions.

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157

2. Experimental requirements for highest resolution For the experimental study of line shapes in angle-resolved photoelectron spectra an excellent instrumental resolution is crucially important. The instrumental energy resolution should be considerably smaller than the intrinsic line width. For example, surface states on noble-metal surfaces are observed with line widths smaller than about 50 meV and the narrowest line observed up to now is measured with 23 meV FWHM. The instrumental energy resolution is determined by both the line width of the exciting UV-radiation and the energy resolution of the electron energy analyzer. Besides energy resolution the angular resolution is of great importance as well. Line shapes of peaks with large angular dispersion are changed considerably when the angular resolution is not sufficient. In several cases a resolution of better than ± l ° is required for studies of surface states. Angular resolution is determined by the characteristics of the electrostatic lens system in front of the analyzer. Due to the fact that the components of the instrument influence each other with respect to the transmission and the resolution power, the design of a spectrometer must be considered carefully in order to get an excellent overall resolution. This will be discussed in Section 2.4.

2.1. Energy resolution function of a hemispherical analyzer Most of the analyzers used in high-resolution photoelectron spectroscopy are hemispherical. In the discussion of their resolution function we will follow a discourse of Sevier [9] upon different analyzer designs. The analyzer accepts an electron beam diverging in two dimensions at the entrance slit, with angle a in the plane of the analyzer and/3 in the perpendicular plane (Fig. 1). The analyzer is focussing the beam in both directions onto the exit slit. Beams with different angles/3 are focussed perfectly, which is the advantage of the spherical design. Beams with different angles c~ are focussed only in first order approximation. Due to higher order terms a monoenergetic point source with a given range of angles - a m < c~ < am results in an image with trace width wt = 2~o~2, where ~ is the radius of the electron beam entering midway between the two spherical electrodes.

4

k

ro w1

D

r180 w2

Fig. 1. Schematic diagram of an electrostatic hemispherical analyzer with mean radius ~, inner sphere radius a, outer sphere radius b, entrance and exit slit width wt and w2, respectively.

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R. Matzdorf/Surface Science Reports 30 (1998) 153-206 I

I

I

w2 > w 1 cO

1.0

13)

Bose width

~

b)

/ \

/ \ //w2=wl\\x

Z

0.5 I,--

/

//f / .. ,'w2 < wl L' , \ \ I

-1.0

0.0

0

1.0

AE/Eo [wi/27 ]

1 w2/wl

2

Fig. 2. (a) Energy resolution function for an analyzer without aberrations ( a = 0) displayed for different cases of slit widths. Solid line: w2 = 1.5Wl, dashed line: w2 = Wl and dotted line: w2 = 0.5Wl. (b) Base width, full width at half maximum (FWHM) and top width of the trapezoid as a function of (from [9]).

w2/wl

The relation between an electron beam with energy E entering at an arbitrary radius r0 and its image point at radius rl8o (compare Fig. 1) is given by r0 2E07 r180 - Ero cos 2 a

1,

(1)

where E0 is the energy on the central orbit with radius ~. E0 is called the pass energy. From this relation the dispersion of the electron beam on the exit slit is derived in first order approximation to D = Eo. Or/~E = 2~ at E----E0 and a - - 0 . We will discuss in the following, first, the resolution function of an analyzer with narrow entrance and exit slits of width Wl and WE, respectively, and afterwards with a large area positional sensitive detector. In the case of vanishing aberration (a ----0) the resolution function, which is a graph versus energy of the fraction of the particles at the entrance slit that are transmitted by the analyzer, has the shape of a trapezoid (Fig. 2(a)). The base width and top width of the trapezoid are plotted in Fig. 2(b) as well as the full width at half maximum (FWHM) of the resolution function, which is identified in most cases with the energy resolution AE. It is given by A E = Eowl/2~ for W2 < W1 and A E = Eow2/2~ for w2 > w]. If the entrance slit is illuminated uniformly, the intensity behind the exit slit is proportional to EoWlwz/2F. Obviously, it is optimal to use Wl = w2 = w for best resolution and intensity at once. By introducing aberrations the resolution function becomes asymmetric due to the fact that aberrations are quadratic in a. Now one could argue that in actual measurements performed at high angular resolution this term is small, but this argument is wrong, since a m is not equal to the maximum angle that is accepted at the entrance of the lens system. The angle am can be considerably larger due to the retardation of the electrons from kinetic to pass energy. In Fig. 3(a) the shape of the resolution function for an analyzer with wa = w2 = w is given for different am in units of x/~-/2~. The triangular shape of the resolution function for am = 0 is changing to one, which becomes more and more asymmetric at larger electron beam divergence. Additionally, the center of the distribution is shifted in energy by S = aZm/3, which can cause problems with the energy scale. The energy resolution is approximately given by

2 AE w am + E0 - 2 7 4 ' as long as the aberrations are not dominant.

(2)

R. Matzdorf/Surface Science Reports 30 (1998) 153-206 I

I ~

I

~ 0

I

~)

159 I

0.05

I

b)

E / E o = 1.05

a 2m = w / 2 f

g

E / E o = 1.0

7:

0.00

0

'~,

I'--

a2 = 2w/27

"'~'/X "~'\ /

a 2" =

4w/2?

E / E o = 0.95 -0.05

1.00

1.01 E/Eo

1.02

I

I

I

-0.05

0.00

0.05

x/27

Fig. 3. (a) Energy resolution function of an analyzer with aberrations olm for the case wl = w2 = w. The function is plotted for different values of C~m.(b) Image of a rectangular entrance slit at the detector for three different electron energies counted simultaneously.

In particular, when an analyzer is optimized to high resolution the concomitant loss in intensity may be compensated by using a position sensitive detector. The narrow exit slit is replaced by a row of rectangular stripes on the detector aligned along the dispersion direction and assigned each to a counting channel. The recorded intensity in a spectrum is proportional to Eowi Wd/2?, where wd is the width of the detector, i.e. the width of all the stripes in sum. In general Wd is considerably larger than w2 and not relevant for the resolution power. The intensity therefore decreases linearly with increasing resolution in contrast to a quadratic decrease in an analyzer with two narrow slits, neglecting aberration effects. Two effects must be spent attention to. Firstly, the image of a rectangular entrance slit is not a rectangle and, secondly, the energy dispersion cannot be treated in the linear approximation across the rather large dimension of the detector. In Fig. 3(b) the image of a rectangular entrance slit is plotted for electrons with three different energies, counted simultaneously on the large area detector. The images are not equidistant due to the non-linear dispersion. To compensate for both effects, any subdivision of the detector into channels must be non-equidistant and the entrance slit must be adjusted in its shape. The quadratic term in the non-linear dispersion relation can be suppressed by using an equipotential Herzog plate (as is the extended detector) instead of fringing field correction plates [10,11].

2.2.

Electrostatic lens systems

The purpose of the electrostatic lens system is, firstly, to focus the electrons onto the analyzer entrance slit; secondly, to retard them from kinetic to pass energy and last but not the least to select only those electrons which are within the solid angle cone of detection. The most common way to realize the first two points is to use one or two stages of cylindric triple-element lenses. A typical so-called transfer lens system is shown in Fig. 4(a). These lens systems designed for XPS and UPS purpose are optimized to image the illuminated area on the sample onto the entrance slit of the analyzer. The angular acceptance cone of these lenses is generally not restricted to a small angle a m.' A restriction can be most

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160

o)

sample

entrance slit analyzer

area V . . . . . . . . . sample

I............ aperture

Fig. 4. (a) Schematic diagram of a two-stage transfer lens system designed of cylindric triple-element lenses. The rays of electrons are indicated schematically by dotted lines. Also indicated are the maximum angles a~mand o~m at the sample and the entrance slit of the analyzer, respectively. (b) Restriction of the acceptance solid angle by an aperture in front of the lens system a~ depending on the diameter of both illuminated area and aperture.

simply achieved by introducing an aperture in front of the lens system. But o~m ' depends on both the diameter of the aperture and that of the illuminated area, respectively (Fig. 4(b)). Angular resolution cannot be chosen below a limit determined by the size of the light spot. The angle t ~ m a t the entrance slit of the analyzer is given by the Helmholtz-Lagrange relation O~ m ~ - O/mV/Ekin/Eo/M with the pass energy E0 and the magnification M of the lens system. M is commonly set to unity, which results in an enhanced angular spread of electrons at the entrance slit of the analyzer as compared to the spread in front of the lens system. Formula (2) for the energy resolution g of the analyzer given in Section 2.1 can be rewritten with o~m as

,'XE = Eo

w

+ Erjn--

r2 1 o~m

4

(3)

As mentioned above, the second term can become relevant for the energy resolution even at very good angular resolution conditions. An optimized design of a transfer lens is discussed by Kevan [12]. However, putting an aperture in front of the lens system is not the best way to restrict the acceptance solid angle. In particular, when a comparatively large area on the sample is illuminated, this method fails. A more successful way is to introduce an additional lens element, in a way that an incident pencil of parallel rays is focussed onto a pinhole with diameter d, see Fig. 5. Electrons with an angle a 't greater than a m,t = d / 2 f are stopped at the pinhole, if f denotes the focal length. The advantages are obvious: firstly, the angular resolution function is step-like with a uniform transmission; secondly, extensive illuminated areas could be used for angle-resolved measurements. Both points are important to achieve maximum intensity in high-resolution measurements. A self-built lens for this purpose was used for many results presented in this report. It was operated with the potentials V1 = 1,'l grounded. This kind of symmetric triple-element lens designed as acceleration lens has comparatively small aberration errors [13]. The additional lens is positioned with the pinhole at the focus of an unchanged commercial lens system as discussed above. By variation of the pinhole diameter d the angular resolution in both dimensions may be tuned down to -4-0.4°. A practical guide for the construction of electrostatic lenses is found in [13].

161

R. Matzdorf/Surface Science Reports 30 (1998) 153-206 Vl ~llrn ~

V2

V3

'

f

Somple

Pinhole

Fig. 5. Schematic diagram of the pre]ens designed as a cylindric triple-element lens. An incident pencil of parallel rays is focussed onto a pinhole (dashed lines). Electrons entering with an angle greater than c~ into the lens are stopped at the pinhole

(dotted line).

The angular spread of the electrons behind the pinhole is determined by a m" and the extension of the illuminated area on the sample. The maximum angle a m ' is given by a m ' ----D/2f+ am, " where D is the diameter of the light spot. The resulting energy resolution of the analyzer can be expressed for this system by w

A E = E0 ~rr + Eking-~

+ a

.

(4)

The focal length of the prelens f should be adjusted in a way that under the desired resolution conditions the pinhole is imaged in its size exactly on the entrance slit of the analyzer (w = Md). Using this optimized condition the energy resolution can be written as w

/Xe=E0 +ek n

(5)

Generally, it is not necessary that the stop for electrons with a " > a "m is located in the first focal plane. Martensson et al. [14] recently suggested an analyzer with a lens system, where the entrance slit of the analyzer is used as a stop in one dimension. In the other dimension the angular resolution is determined in the last focal plane, the plane on the detector, by selective counting. In this system aberrations due to the retardation lens and the analyzer must be taken into consideration with respect to the angular resolution, additionally. 2.3.

Light sources

Not only the analyzer energy resolution, but also the line width of the light source is, of course, relevant for the overall instrumental resolution. Using synchrotron radiation, the energetic line width is determined by the monochromator resolution. However, we will restrict our discussion to laboratory UV-light sources. In a discharge lamp line radiation is produced by electronic transitions between energy levels in neutral atoms and ions. Usable photon energies of the rare-gases are 21.22eV (HeI) and 40.81 eV (Hell) as well as the line doublets 16.85/16.67 (NeI), 11.83/11.62eV (ArI), 10.64/10.03 (KrI) and

162

R. Matzdorf/Surface Science Reports 30 (1998) 153-206 Helt line at 21.218eV HOi~W cathode source 58-'.33A

80 60 /,0

20

0

-20-40 "60 '1"

ECR Source

58/, 33A

0

-20",~O-ti0

Fig. 6. The Helc~ line emiUed from a hollow cathode source (left) and the ECR UV source (right) as recorded by a 3 m normal incidence grating spectrometer at a resolution of a few tenths of a meV (from [10]). Note that the intensity of the ECR line is about 30 times the intensity of the hollow cathode line [10].

9.57/8.44 (XeI) [15]. In radiation from He discharges two mechanisms mainly contribute to the line width, self-absorption and Doppler broadening caused by the motion of the atoms or ions in the radiating plasma [10,16]. The lifetime broadening of HeI-radiation and all other broadening mechanisms are at a level below 10 ~teV despite of fine structure. The Hell-radiation is a spin-orbit split doublet with a peak separation of 0.7meV [10]. The Doppler effect gives rise to a Gaussian distribution with an FWHM of A E = 7.2 x 10-7x/@--/M • E, where T is the absolute temperature in kelvin and M the mass number [10]. Due to this effect the line width of HeI-radiation emitted at T = 1000K is 0.24meV. Self-absorption leads to considerably larger line broadening. The scattering cross section of the unexcited gas atoms is extremely large for the resonance radiation. This leads to a strong quenching in intensity in the center of the Doppler line profile and only the extreme wings of the initial line profile remain. In Fig. 6 measured line profiles are plotted for HeI radiation emitted from a hollow cathode source and a microwave excited discharge source (ECR source), respectively. Self-absorption is depending on the gas pressure, and for capillary discharge sources the high-resolution limit is reached at low pressure with a substantial reduction of intensity [10]. At the intensity maximum versus pressure of a self-built discharge lamp used by the author a line width of about 15 meV was measured [65]. Besides the first resonance line HeIa (21.22eV) other lines are emitted with small intensity like HeI/3 (23.09 eV) and HeI7 (23.74 eV). HeI/3 has about 2% and HeI'7 0.5% intensity of the HeIa-line at gas-pressure conditions optimized to highest intensity [ 1,15]. By reducing the gas pressure the relative intensity of HeI/3 is rising considerably. The weakness of the HeI/3 line relative to HeIa and its decrease with pressure can probably be explained by the fact, that a collision process takes place before the excited 31P state can radiate [17]. So we realize that low gas pressure is required for a narrow resonance line, and on the other hand at low pressure the relative intensity of HeI/3 is unacceptably large for most photoemission experiments. The only solution of this problem is to use a monochromator.

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163

2.4. Design of an optimized system In the discussion of a high-resolution system consisting of a spherical analyzer with an electrostatic lens system and a UV-source, the relation between the spot size of UV-radiation on the sample and the entrance-slit size of the analyzer is of particular interest. A system works best when all electrons with energy Ekin emitted into the solid angle of interest are counted at the detector, independent of the spatial point of emission. Using a transfer lens system this is fulfilled when the spot on the sample is imaged on the entrance slit in its size according to the relation D • M <_ w [ 12]. If the illuminated area is larger, transmission is reduced since electrons are stopped at the slit. By the use of a lens system that focusses a pencil of parallel rays in a point on the entrance slit of the analyzer, the image size on the slit is independent of D. In this case the angular spread at the entrance slit is depending on D, which limits the energy resolution according to Eq. (5). By introducing an additional slit pair at the entrance slit in order to restrict C~m, the spatial extent of the area on the sample where electrons are counted from is reduced correspondingly. This leads to a loss of transmission as well. However, we must distinguish between the two spatial dimensions parallel (x) and perpendicular (y) to the dispersion direction of the analyzer. The entrance slit of the analyzer may be long in y-direction as well as the angular spread/3m may be large in this direction, due to the imaging conditions of the analyzer. The spatial extent of the light spot in y-direction can therefore be large with respect to energy resolution. If a prelens is used, angular resolution is not affected by D along y. The restriction to D . M _~ w, which is a good choice for both lens designs, is particularly required in x-direction. There are two solutions for this requirement. The first one is to adjust the dimensions of the analyzer to the spot size. This can become extremely cost-intensive. The second one is to focus the UV-light at least in one direction. The radius of the analyzer can be reduced considerably in this case without loss of transmission or resolution power. Focussing can be done for example by a toroidal mirror positioned behind the monochromator of the UV-source. Some experiences with an ellipsoidal mirror are discussed in [ 18,19]. In Fig. 7 a design of a toroidal grating monochromator with subsequent focussing is suggested. The UV-source of a radiating plasma has a typical diameter of 2 ram. It must be reduced in size by a factor of 10 to fit to a typical entrance-slit width of some tenths of a millimeter. Due to the fact that focussing is required only toroidal mirror

// / j

UV

-

i

/'~ t

I-

I I

I

sample

I

source

/ / ' ~

-. "" ~

--\

/ -.

/

/ /

f/

""

Exit pinhole of monochromator

Toroidal grating

Fig. 7. Schematic diagram of a toroidal grating monochromator with subsequent focussing of the UV-radiation by a toroidal mirror. The monochromator is required for suppression of the satellite emission lines.

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R. Matzdorf/Surface Science Reports 30 (1998) 153-206

in one dimension, the tHroidal mirror can often be replaced by a spherical one. The focus is then elongated by the astigmatism of the mirror. Its advantage is the low-cost realizability. If polarized light is required for the experiments, the focussing element can be integrated in the polarizer as one of the reflecting surfaces.

3.

3.1.

Line shape of bulk-emission peaks

The photoemission process

In the one-electron picture the photoemission process is described by the interaction of an electron with the time-dependent electromagnetic wave field A(r, t). This is expressed in the non-relativistic case by Schrrdinger's equation ] ih~O b(. r , t )=[ ~ m ( ~ e V - - A ( r , t ) ) 2+V(r) ~(r,t)=H~b(r,t), c

(6)

where ~b(r, t) is the wave function of the electron and V(r) the unperturbed potential in the solid. The Hamiltonian H can be split in the unperturbed part H0 and the Hamiltonian for the interaction Hin t h2 Ho~b 2m A~ + V(r)~b, (7) e2 ]

1 [-2ehA.V~-e.h(divA).¢+ ic lC

HintS3 = ~mm [

C'2

IAI2~ "

The last term of Hin t Can be neglected in the case of small electromagnetic field strength. It becomes relevant in photoemission experiments with high-intensity laser pulses. The term with divA cannot be suppressed by choice of gauge due to the presence of the surface potential, which breaks the translation symmetry [20,21]. This term is small inside the crystal since A is varying only slowly in space, but it can become relevant in the surface [21]. In a very recent investigation on Ag(1 1 1) the asymmetry of a direct-transition photoemission peak from the sp-valence band has been interpreted by this surface emission [22]. We will neglect the effect in the following description of photoemission, but will come back to it later in Section 3.9. In first order time-dependent perturbation theory the transition rate between an initial state li) and a final state (f[ is given by Fermi's golden rule 27r wfi ~ --h--[(flH~ntli)[26(Ef- E i - hw) (8) with the remaining interaction Harniltonian n~tnt - -

eli imcA "

V --

e mc A

"p,

(9)

Ei and Ef are the eigenvalues of H0 for initial and final state, respectively, and hw is the photon energy. Eq. (8) is valid only for a perturbation acting on the system for an infinite time t ~ He. If the coherent time t of the radiation is limited and the condition t >> 27r/~; is fulfilled, we get 27r wfi : T I(fln~ntli)]2Pf = IMnI2Qf'

(10)

R• Matzdorf / Surface Science Reports 30 (1998) 153-206

165

where Of is the density of states at Ef. This is the transition rate into an energy interval around El, which is large compared to the uncertainty in energy of the photon• If the energy distribution of electrons emitted from one initial state with well-defined sharp energy is measured, its width is determined by the uncertainty in energy of the radiation, due to its limited coherent time. Moreover, we must consider the conservation of wave vectors in the photoemission process. In the relationship kf -- ki -+-kphotonq- G with the wave vectors of the initial and final state ki, kf, respectively, and a reciprocal lattice vector G of the bulk, the wave vector of the photon Ikphoton[ is commonly neglected in the UPS regime. It is 0.01 ~-1 for a 20eV photon and therefore smaller than 1% of a common bulk Brillouin zone dimension. Neglecting kphoton is equivalent with the assumption of a spatially constant vector potential A. With this assumption the matrix element simplifies to iMf l 2

27re2

-- lim2c2 [A. (flpli)

12

27re2

lim2c2 IA .Phi 2,

(11)

where Pf~ is the so-called momentum matrix element [23,82]• Separating out the k-conservation from Mn, which yields/f4n [5], the photoemission intensity inside the crystal can also be written as

I(Ee)

I tf l 2.

~(Ef - Ei - h~v). 6(kf - ki - G).

(12)

i,f,G

When the electron transmits the surface, the wave vector kll parallel to the surface is conserved [24] modulo gll, a vector of the surface reciprocal lattice, according to kvlL -~ kf H+ gll" In this equation kvll is measured in vacuum, i.e. outside the crystal. The correspondence with the electron emission angle 0 with respect to the surface normal is then given by

Ikv111

sin0•

=

(13)

The energy and angle-resolved detection may formally be included by the Kronecker-symbols ~(Ekin- Ef q-~) and ~(kvll- kfll- gll)' where • is the work function. The photoemission intensity measured in the detector is then given by [5]

Iv(Ekin,kv[I) ~

~

I~1fi(kf,ki)126(Ef - Ei - hw) . ~ ( k f - k i - G )

i,f,G, gll

• 6(Ekin -- Ef + ~b)-~(kvll -kfll -gll)"

(14)

In case of finite resolution, i.e. with respect to AEkin and A0 as well as to Ahoy, the Kronecker symbols will have to be replaced by the relevant distribution functions. This will be discussed further below•

3.2.

Lifetimes of photoelectron and photohole

In Section 3.1 we have discussed the photoemission process in the single particle picture. But the solid is a complicated many-body system of electrons all interacting with one another [26]. In order to cope with this problem it is very helpful to introduce quasiparticle states for the screened and decaying hole left behind and for the photoelectron [27,28]. The quasiparticle wave functions satisfy Schr6dinger's single particle equation containing the self-energy Z, which is complex and energy dependent. Its real part describes the screening, shifting the energy from the Hartree value, and the imaginary part describes the decay or inelastic scattering processes [26].

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166

We must be aware of the fact that the real part of the self-energy is not accessible in a photoemission experiment. The measured energy is the quasiparticle energy of the hole left behind and not the single electron in ground state. In the following we will treat this experimental energy as initial-state energy. The imaginary part of the self-energy gives the initial state a finite energy width• Assuming an exponential decay in time the spectral function ,Ah (E) of the hole is given by Ah = (Ef - E i - ho3) 2 -~- F 2'

(15)

where Fh = [ Im{S}l and Ei is the initial-state energy including the real part of S [26,28]. The lifetime of the photoelectron results in quite different effects [7], because in the angle- and energy-resolved experiment scattered and unscattered photoelectrons can be distinguished• The photoelectron can interact with other electrons and also with phonons. In this section we exclude the electron-phonon interaction, which will be discussed in Section 3.7. The scattering of the photoelectron with electrons (electron-hole pair excitation) or the creation of plasmons is always correlated with a drastic loss in energy. These scattered photoelectrons ("secondary electrons") are found in a more or less continuous spectrum, which is generally treated as a background [31]. The unscattered photoelectrons are found in the photoemission peaks we are interested in. They come out of a region near the surface with a probability, which is damped exponentially into the surface. This limited escaping depth is responsible for the surface sensitivity of photoelectron spectroscopy. The photoelectron final state may therefore be treated as an exponentially damped wave function, a so-called inverted LEED state [25,26]. Nevertheless, the final-state energy of the unscattered electrons is measured in the vacuum much sharper than expected from the lifetime uncertainty of the final state: Not the uncertainty in energy but the uncertainty of the final-state wave vector in surface normal direction is relevant for the photoemission process. The exponential decay in space corresponds with a Lorentzian distribution of wave vectors perpendicular to the surface kf± around the wave vector k°± of the undamped final state /~(kf±) =

00f/71(kf± - k°±) 2 q- 0°2.

(16)

The width of the distribution O'f can be calculated from the imaginary part of the self-energy via the group-velocity of the final state 00f = Ok±/~Ef.Im{Sf} assuming a linear band dispersion. Let us now calculate a photoemission intensity by substituting the &function in Eq. (14) with the corresponding distributions:

Iv(Ekin,k°[i) (x ~ I/~/fi(kf,ki)] 2. (.ahw • .Ah)(Er - Ei - ho3)./Z(kf± - ki± - a ± ) f,i,G,gll •/~(kf[I - kill - GII)" 79e(Ekin - Ef + ~ ) . 79k(kvll - kflI - gJI)"

(17)

In this formula .Ah and/2 are defined by Eqs. (15) and (16), respectively..AM is the spectral distribution of the light source, which has to be considered only if significant with respect to the experimental resolution. The analyzer resolution function, approximated for example by a Gaussian, 1 ( 1 (Ekin - Ef - ~)2- ) DE -- v,,~Fa~texp - 2 ?2t

(18)

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167

replaces 6 ( E k i n - E f + eb). The parameters Fde t and the experimentally determined AE (FWHM) are connected by A E = x/8 ln2/'det. The angular resolution is accounted for by introducing a twodimensional acceptance function Dk(kvll) for an area in kvll, which is correlated to the detector solid angle. The wave vector conservation is split in the conservation of the k i and the kll component of kf, ki and G, respectively, .Ar~ * .Ah is the convolution of both distribution functions. For practical use let us rewrite the sum over discrete states as integrals in E and k over continuously dispersing states:

G,gLI

x E ( k f ± - ki± - G±) • DE(Ekin -- Ef + q~)

x D~(kvll - kill - GIt - gll) " dkf± dkill d e f .

(19)

To calculate with this formula the photoemission intensity requires the band structure Ei (ki) and Ef(kf), the matrix element [2(4n(kf,ki)] 2, the angular and energy resolution functions DE, Dk and Fh(Ei), o-f(kf±) and Fh~. In this formulation only one initial-state band is used. To account for the six initialstate valence bands of the transition metals it must be summed up over the band index j = 1 . . . . ,6, using the band dispersions E~(k). One approximation is very helpful for practical use of this formula. The matrix element is assumed to be independent of k within the uncertainty of k± and within the experimental resolution. For an analysis of experimental data ,('/fi, /'h (E) and crf (E) may then be treated as parameters, which are determined from experiment. As input a calculated band structure and the experimental resolution functions are used. 3.3.

E x p e r i m e n t s on a l o w - i n d e x surface

In this section experiments on Cu(1 1 1) will be compared with calculated photoemission intensities obtained by a simple method of k±-integration. The experiments were performed in an energy range where only one reciprocal lattice vector couples to a final state which allows escape into vacuum, and therefore no umklapp-processes are possible. In Fig. 8 energy distribution curves (EDCs) are shown measured at different electron emission angles 0 in the F L U X mirror plane of the bulk Brillouin zone. The sample temperature was T = 100 K, a temperature where electron-phonon interactions are of minor importance. The experimental energy resolution was set to 3 0 m e V with an experimentally verified Gaussian distribution, including the energetic width of the photon source, which was about 18 meV in this case. The angular resolution of +0.7 ° was realized by using the prelens described in Section 2. The angular resolution is good enough to neglect it in a calculation of line shapes of bulk-emission peaks. Therefore in Eq. (19) the integration over kvll is not necessary and the integration over Ef can be rewritten in a convolution t

0

Iv(Etin,k°ll) = (.Ah~ * 79E) * I'v(Ekin,kvll)

with l'v(Ekin, kvll) , o oc IffIfi(kf,ki)l 2 f .Ah(Ef - Ei - h w ) - £ ( k f ± - ki± - G±)dkf±,

(20)

0 using the relationships kill = kvll, GII = 0, glt = 0. This equation can be used very effectively to model the line shape of bulk-emission peaks and is now used to calculate the shape of the Cu(l l l) spectra. As mentioned above we need an initial- and final-state band structure. The initial states are calculated with a

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relativistic Korringa-Kohn-Rostocker (KKR) band-structure code supplied by Braun and coworkers [32]. Calculations were performed in the FKLUX-mirror plane on a two-dimensional lattice in k-space and were tabulated with about 3000 points. For integration the bands were interpolated with a twodimensional cubic spline. The energy dependence of Fh is chosen quadratic in the energy distance from the Fermi energy EF. It fits best to experimental data taking Fh = 0.006 e V -1 (Ei - EF) + 0.01 eV. The final state is assumed to be free-electron-like as Ef----h21kfl2/2m*+ Vo, with an inner potential V0 = - 6 eV with respect to EF, and an effective electron mass m* -----m equal to the electron rest mass. The energetic lifetime width of the final state is taken to be Ff = 0.065(Ef - EF) in accordance to data from several authors collected in [33]. The width in wave vector (re is calculated from the final-state band dispersion assuming (re = ~k±/~Ef. Ff. The integration and subsequent convolution is done numerically, which runs for less than 1 min on a modem workstation. The integration range is

"'---'T

f

I

1

Cu(111) he=21.2eV T = 100K FLUX

o--oo

L[

o

Cu(111) h~o=21.2eV T=

L

look

FLUX

A1,

I

I

F---

(9=5 °

(9

=

20 °

e=

®

=

25 °

E 4-, E

.,.,x E

10 o

•••

•

iJ ___:"

sot

', E I

-6

(1)

-4 -2 Initiel stGte energy [eV]

0

-6

(2)

I

-4 -2 Initiol stote energy [eV]

0

Fig. 8. Electron energy distribution curves taken from Cu(1 1 1) at different electron emission angles 0 in the FLUX-mirror plane of the bulk Brillouin zone. Experimental data are reproduced as dots. Solid line: shape of spectra calculated by k~integration, for details see text. Dashed line: secondary electron background fitted as polynomial of second order.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

Cu(t11)

I'

]

169

I"-

h~=21.2eV I-

T = lOOK FLUX

i Et.

c

$2

0 = 40 °

r-

C

/

S2

@ = 50 °

c

/

S2

c

-6 (3)

-4

-2

Initial state e n e r g y [eV] Fig. 8

(continued).

displayed graphically in Fig. 9 in the extended zone scheme of the bulk Brillouin zone. The results are shown in Fig. 8 as solid lines. We find in all spectra a good overall agreement in the shape of the peaks. This demonstrates the efficiency of our comparatively simple method of integration. However, we see some deviations from the experimental line shapes. First of all, the emission out of the surface states (labeled Sl, $2 in Fig. 8) is not included due to the fact that a bulk band structure is used. Secondly, weak intensity due to the HeI satellite radiation at ha; = 23.1 eV (labeled sat) is not included in the calculation. Also, energetic peak positions are reproduced in some cases not perfectly, which is explained by deviations in the initial-state band structure and the free-electron:like final-state band. This is not surprising since the real part of the self-energy is not included in the "ground state" band structure. The most obvious difference between experimental and calculated spectra is additional photoemission intensity (label D, E) found in the energy range Ei = - 3 . 5 , . . . , - 2 . 5 eV of the experimental data. It is often called "density-of-states transition intensity" [1-5] and is observed especially in the normal-emission spectrum (label D) [29]. Perhaps, this intensity is due to surface emission, but another very probable explanation in the context

R. Matzdorf /Surface Science Reports 30 (1998) 153-206

170

I I

I

k, kl'j

X

c I 'x~

Ikl

Fig. 9. Extended zone scheme of the FLUX-mirror plane in the bulk Brillouin zone of Cu(1 l 1). Inner circle: free-electronlike final state corresponding to an initial-state energy Ei = - 6 eV exited with h~ = 21.2 eV; outer circle: corresponding to Ei = 0 eV. Also shown are kll and kll for these two energies in a spectrum taken at 0 = 50 °. The integration over k± runs along the vertical lines with kll depending on Ei. The center of the distribution/~ is marked by open squares for both energies.

I

I

I

I

13

T = lOOK

~

o=soo

I

1 11

/It\

A i,i: ii

....

-6

I

I

Cu(111) h~=21.2eV

•

I

-5

~"~::"

......

--

--'5 X.,,..,.#.

-4 -3 -2 lnitiol stete energy [eV]

.__

-1

0

Fig. 10. EDC taken from Cu(1 1 l) at 0 = 50 °. Experimental data are plotted as circles. Solid line: shape of the spectrum calculated by k±-integration. Dashed-dotted line: secondary electron background. Dotted lines: peak shape of two emission features originating from the initial bands with band index 1 and 3, respectively. Inset: peak shape calculated with (upper) and without (lower) including angular resolution.

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171

of Sections 3.1 and 3.2 is to relate it to our assumption that IMnl2 is independent of k±. In Fig. 10, which is a magnification of the 0 = 50 ° spectrum in Fig. 8, the line shape of two emission features is plotted separately. Not only do we recognize that the peak shapes may be significantly different and non-Lorentzian-like, we also see that the line shape of peak B would fit better to experiment if the matrix element would be allowed to increase on the right-hand side as compared to the left-hand side. Fig. 10 additionally displays the effect due to finite angular resolution (see inset). It was calculated according to

IIv( Ekin , k°ll )

0(

I](,/fi(ki)12 _I .i~h(Ekin -- E i - l i w -

~)'/Z(kf±-

• 7)11(kvll - kill - GII) dkf± dki]l •

ki±- G±) (21)

The effect on line shape is not dramatic around Ei = - 1 eV and is invisible in the rest of the spectrum, but it can become relevant of course in spectra recorded with angular resolution worse than the assumed +0.7 ° . The main advantage of the data analysis presented in this section is that it allows for a clear and comparatively rapid identification of the influence of Fh and of on the experimental spectra. Turning the argument around: the results reproduced in Figs. 8 and 10 clearly demonstrate that the line shapes of bulk-emission peaks at Ei < - 2 e V from Cu(1 1 1) are determined almost exclusively by Fh and af. This situation is in clear contrast to the surface state emission discussed further below in Section 4. Another great advantage of this simple method of k±-integration is that photoemission intensity due to umklapp processes or transitions into additional final-state bands can be identified reliably. Last but not least, the method is very helpful to reveal the energy dependences of Fh and af which are required as input data in one-step calculations (see Section 3.6). In an iterative determination we need about a minute of computer time for one iteration, whereas a single one-step calculation needs several hours. In the past several authors derived an analytical formula for the measured line width F m using the approximation of linearly dispersing bands not only for the final, but also for the initial state [7,26,34 43]. If the spectral function of both states is a Lorentzian, the measured line shape is as well within this approximation. The experimental line width is then given as

ff'm z

Fi/lViL ] -'}-Ff/IVfL [

(22)

[1 - (mvill sin 2 0)/hkll ] -A_yf±[l -- (mvfl[ sin 2 0)/~kl[] w h e r e viii, Vfll, vi± and vf± are group velocities (hVil I ~- OEi/~kll and so forth) and F f is the imaginary part of the spectral function in energy of the final state. A detailed derivation is published e.g. by Smith et al. [44]. This formula is adequate in all cases where the dispersion of the initial state is linear. However, as demonstrated by "peak" B in Fig. 10, severe deviations from Lorentzian line shapes must be expected, if the assumption of a linear initial- and final-state dispersion is violated in particular k-space regions. Nevertheless, Eq. (22) has been applied sucessfully in several earlier [36,37,41,42] studies as well as in more recent investigations [8,45,46] employing better instrumental resolution.

3.4.

Experiments on a vicinal surface

On perfectly ordered low-index noble-metal surfaces no umklapp processes can occur below about 24 eV photon energy, but scattering at randomly distributed steps on "real" surfaces is allowed. An

172

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

git= 0,572~-1 [ 3a

X

I\

F2°° /t;"

k//

"

X .,/W

X

F22o,

l

T

]

W X,

r00

Fig. 11. Extended zone scheme of the FXWK-mirror plane in the bulk Brillouin zone of copper. The stepped surface of a Cu(6 1 0) crystal is indicated in real space. Directions of k i , kll and the reciprocal lattice vectors of the surface gll are given. The flee-electron-like final states, which are relevant for spectra given in Fig. 12, are plotted as circle segments for Ei = - 6 eV (inner circle) and Ei - - 0 eV (outer circle), respectively.

additional broadening mechanism in photoemission spectra results from these scattered electrons. A quantitative incorporation in a calculation can be realized by a summation over all possible gll scattering vectors weighted by the form-factor of a one-atomic step edge. In order to estimate the formfactor it is of interest to look for umklapp processes on an almost regularly stepped vicinal surface. Experiments were performed on a C u ( 6 1 0 ) surface in comparison to a C u ( 1 0 0 ) fiat surface. The C u ( 6 1 0 ) surface is cut at an angle of 9.5 ° with respect to the crystal axis. The resulting mean terrace width is 3a with the lattice constant a. The step edges are oriented along the [001] direction. Fig. 11 shows the FXWK-mirror plane of the bulk Brillouin zone and schematically also the surface in real space. The reciprocal lattice-vector gll is oriented parallel to the macroscopic surface and not parallel to the terraces, the kf_L direction is perpendicular to the macroscopic surface as indicated in Fig. 11. Due to the fact that the final state is damped along k±, the integration over kf± in a calculation of line shapes runs along this direction in contrast to a Cu(100) surface, where it runs parallel to the crystal axis. The wave vector of a free-electron-like final state achieved with ha; ----21.2 eV is located between the two indicated circle segments. In a calculation for a normal-emission spectrum kf± integration runs along the solid line in Fig. 11 for the direct transitions and along the dashed lines for umklapp transitions.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

CuOoo)

o)

he=21.2eV : T = 125K FXWK

o--loo

c,,(6~o)

I J

0=2'5° T = 308K i

11 ~

b)

i

0 = 22 °

:}

.

i

i ~'

c-boo)

d)

E ) = 15 ° >.,

Cu(61o) e = -2,5 °

I:

e)

Cu(61o)

f)

(9 = 28 °

;i

i :

/

c.(,oo!

g)

f

-4-

i:

: !:

.

i

e=2o

-6

c)

cu(61e) T -- 3o

/

173

c.(61 o)

h)

e = -7.3 °

-2

0

-6

-4

-2

0

Cu(610) 0 = 33.2 °

-6

-4

,~

i)

-2

0

[nitiol s t a t e energy [eV]

Fig. 12. Electron energy distribution curves taken from Cu(1 0 0 ) (left column) and Cu(6 1 0) (middle and right column). Three spectra displayed side by side were taken at electron emission angles corresponding all to an equivalent k-point in the Brillouin zone, respectively. Dots: experimental data; dashed line: secondary background fitted as polynomial; solid line: result of kj_-integration calculated with same matrix elements for corresponding spectra. Note that spectra from Cu(1 0 0) were taken at sample temperature T --- 125 K whereas spectra from Cu(6 1 0) were taken at T = 308 K.

In Fig. 12 experimental data are reproduced for both the low-index and the vicinal surface, recorded in the FXWK-mirror plane. The final state inside the crystal is independent of the surface orientation, so one can find electron emission angles on both crystals, which correspond to approximately the same or an equivalent k-space point kf. (This can be achieved exactly only for one energy in a spectrum.) In this sense the spectra 12(a)-(c) are corresponding to one k-space point. It is equivalent for 12(b) and (c) due to the mirror-symmetry inside the crystal. In the same way are 12(d), 12(e), 12(f) and 12(g), 12(h), 12(i) corresponding to each other, respectively. We recognized that no additional structures are exhibited on the (6 1 0) surface in comparison with corresponding spectra from Cu(1 00). In other words, none of the energetically allowed umklapp processes occur with sufficient intensity to be relevant in the photoemission spectra. In conclusion, the form-factor for scattering of a final-state electron (Ef--~ 18eV) at a copper step-edge is quite small. We therefore expect that randomly distributed steps on a non-perfect single-crystal surface do not change spectra of bulk-emission peaks considerably. In contrast, intense adsorbate-induced umklapp processes are observed for example in Cu(1 00)c(2 x 2)C1 [47] and ( v ~ × v ~ ) R 3 0 ° overlayers of Xe and CO on Cu(1 1 1) [205].

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

174

On the other hand, there is an interesting difference between corresponding spectra in Fig. 12 with respect to relative peak amplitudes. Three different explanations can be taken into consideration: first, different weighting of matrix elements due to polarization effects of the light (described by Fresnel equations); second, different surface-transmission coefficients in the sense of the three-step model and; third, changes due to the different directions of kf±. We can easily check for the last effect with the method of kf±-integration. The spectra of Cu(1 00) were calculated using Eq. (20) with the same parameters as used for Cu(1 1 1) and the matrix elements were adjusted to the measured peak intensities. These matrix elements were then used without further adjustment in the calculation of the corresponding spectra on Cu(6 1 0). The different relative peak amplitudes found in the experiment are reproduced rather good. Due to the different direction of kf±-integration line shape and width are changed, resulting in different amplitudes at constant line area. Therefore, we can conclude that the other two effects mentioned above are of minor importance.

3.5.

Final-state band-structure effects

In our discussion of lifetime broadening effects so far we did not spend much attention on the electron final-state band structure. Until now we treated the empty bands only as "free-electron-like", characterized by an empirical value of the inverse lifetime width Ff. In this picture Ff and af are simply related by the band velocity according to /-'f z IOEf/Ok±[. Of. If we approach an energy gap within Ef (k), additional damping occurs which tends to increase O'f considerably. Moreover, crf is temperature dependent due to the modified reflectance of the lattice at elevated temperature. In the present section we will discuss both these lifetime and temperature effects within the model of a complex band structure. This model is extensively described in the book of Pendry [25]. We follow his idea to discuss the effects first in a one-dimensional model, before we present in Section 3.6 the results of a full-relativistic three-dimensional one-step calculation. Our one-dimensional solid is assumed to be a stack of identical layers ending with a surface. An electron moving in this solid perpendicular to the layers is scattered at each layer, characterized by its transmission coefficient t and reflection coefficient r = i v / 1 - t 2 for the amplitude of the wave function. In [25] analytic equations for the relation k(E) are derived by analyzing the amplitudes of forward and backward traveling waves. This is done within a multiple-scattering theory using the kinematic condition of weak scattering. The relevant relations are given [25] by k = .1 ln/3,

(23)

lC

with the abbreviations /3 =

(1 + o~2) -1- V/(1 + oL2) 2 -- 4tx2t2 2t~t ,

oe = exp(ik0a),

k0 = v/2E - 2V0,

(24)

where V0 is the complex inner potential and a is the interlayer spacing. Note that (exclusively in this section) we use atomic units to remain in accordance with [25] for convenience. The lifetime r is introduced in this model by an imaginary part of the potential Im{V0} = 1/27resulting in a complex E(k) relation. If both Im{V0} and r are zero, i.e. infinite lifetime and vanishing

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

175

,-'~2 ,ID

I E~

"'

1

-1

0 Relk I [a.u.]

1 -1

0 1 - 0 . 2 0.0 0.2 Relk I [a.u.] Imtk I [a.u.]

-1

0

1 - 0 . 2 0.0 0.2 Relk I [a.u.] lmlk I [a.u.]

Fig. 13. (a) Free-electron-like final-state band structure in the reduced zone scheme of the one-dimensional model described in text. The lattice constant was chosen as 7r/a = l (a.u.) -I . (b) Real part (left-hand side) and imaginary part (right-hand side) of the complex one-dimensional band structure calculated with parameters: t = 0.9, Im{ V0} = 0. Solid line: T = 0, i.e. perfect geometrical order; dashed line: elevated temperature, i.e. thermal displacement was chosen to be U 2 = 0.1 (a.u.) 2. (c) Same as in (b) but with Im{ V0} = 0.15.

reflection, the relation simplifies to the free-electron-like final state. Fig. 13(a) reproduces the corresponding E(k) in the reduced zone scheme. In this case k is real for all energies. Introducing a finite reflectance of a layer (t -- 0.9) results in band gaps at the zone boundaries, well known from solid state text books. In Fig. 13(b) real and imaginary parts of k are plotted as solid lines. In the band gaps damped wave functions are allowed at the surface. They are damped exponentially in space with an attenuation length determined by the imaginary part of k. If V0 gets an imaginary component, k becomes complex for all energies. In Fig. 13(c) we show the band structure for Im{V0} -- 0.15 by solid lines. No clearly defined band gaps remain in their place and we see crossings of two branches of Re{k}. The band dispersion is more similar to the free-electron-like final state than to the band structure with infinite lifetime. This is indeed an explanation for the fact that a free-electron-like final state is so successful in the kinematic analysis of photoemission spectra. The result for Im{k} demonstrates the contribution of both effects, the reflection in band gaps and the "absorption" by finite lifetime. It is the absolute value of Im{k}, which is equal to ~rf and which must be used for calculation of line shapes as discussed in Section 3.2. For the case of a transition into a band gap crf is enlarged by scattering effects in the bulk. The line shape of the corresponding peak is not increased very much because initial-state bands are fiat at the zone boundaries. However, we have the possibility to calculate o-f within this one-dimensional model, when the lifetime is given. It can, of course, be energy dependent.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

176

Not only the lifetime shows up in the complex final-state band structure, but also temperature effects may change the slope of E ( k ) . We will now discuss effects by weakened reflection in the solid due to reduced geometrical order. This can easily be incorporated into Pendry's one-dimensional model. The reflectance of a layer is reduced by the Debye-Waller factor r =

(25)

roe -1/2Ak2U~,

where Ak = 21k I is the difference in wave vectors of the incoming and the reflected electron and U~ is the square of the amplitude of thermal vibration of an atom about its equilibrium position. U2 will be discussed in more detail in Section 3.7. (Notice a factor 2 in the exponent of Eq. (25), which is different for reflection of amplitude or intensity, respectively.) Introducing the enhanced transmission coefficient t = v/1 - r 2 in Eq. (24) we get the complex band structure at elevated temperature. For this calculation we assume U~ = 0.1 (a.u.) 2, which is typical for noble metals at a few hundred °C [49]. The complex band structure at T > 0 is plotted in Fig. 13 as dotted lines. In the case with infinite lifetime (b) the band gaps become smaller due to the reduced reflectance of the crystal. This effect increases with increasing energy because Ak is energy dependent. The reduced imaginary part of k in the band gaps is a consequence of reduced reflectance as well, because the wave function is damped less, as expressed by a smaller Ira{k}. In the case of finite lifetime and elevated temperature (c) Re{k} is changed only marginally. The imaginary part is reduced due to smaller reflectivity of the crystal, particularly in the band gaps. In the other energy ranges the temperature-independent electron lifetime is dominant. Not only the band structure, but also the reflectance of the crystal may be calculated within the onedimensional model including multiple scattering [25]. The reflected intensity of a plane wave moving into the crystal is given by + "7) + k(1 - "7)_2 ~ with "7 - fit-o~ / ( l - - ,r I ----I0" \k'(1 +"7) --k(1 - "7),]

(

k'=v/~.

(26)

The result is shown in Fig. 14 for three cases. Firstly, with infinite lifetime the reflectance is equal to unity within the band gaps as expected. Secondly, with finite lifetime the reflectance is reduced due to the "absorption" inside the solid and, thirdly, at elevated temperature the reflectance of the crystal is reduced additionally. In a three-dimensional solid the photoemission final state is a LEED-state, as well. It is damped by lifetime effects perpendicular to the surface, but not parallel to the surface. Perpendicular to the surface

I

n.-

I

I

\.

0

I

I

0.5

1.0

~. -° . . . . .

1.5

2.0

I

-.

"~'~

2.5

,

Energy [Hartree] Fig. 14. Reflectanceof the bulk calculatedwithin the one-dimensionalmultiple-scatteringformalism.Parameterswere taken the same as in Fig. 13. Solid line: Im{ Vo} = 0; dashed line: Im{V0} = 0.15 with perfect order; dotted line: Im{V0} = 0.15 but with thermal disorder of U0 2 = 0.1 (a.u.)2.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

177

the final state is quite good approximated by a free-electron-like final-state paraboloid. Parallel to the surface the wave function cannot be damped, evidently, and band gaps are still remaining. These gaps may be affected by temperature as visualized in Fig. 13(b). All these scattering and temperature effects are incorporated in the one-step formalism, which will be presented in Section 3.6. This formalism is based on a three-dimensional multiple-scattering formalism. For an incorporation of temperature effects, the scattering matrix is multiplied with the Debye-Waller factor very similar to the one-dimensional model. Additional effects caused by electron-phonon scattering of the final-state electron will be discussed in Section 3.7. 3.6.

One-step calculations in the Green's function formalism

In the one-dimensional model of Section 3.5, scattering between layers was accounted for, exclusively. In a three-dimensional solid scattering within each layer becomes relevant. In the one-step formalism developed by Pendry [50] the solid is grouped in a stack of layers with a muffin tin potential. The potential acts both as source of photocurrent and subsequent as a scatterer. This three-dimensional multiple-scattering process is divided into inter-layer (between layers) and intra-layer scattering (within a layer) [50]. Lifetime of hole and final-state electron are incorporated by an imaginary part of the potential. The final-state wave function is calculated by an electron plane wave propagating into the solid, while it is scattered in the bulk. Additional final-state bands are transferred into the first Brillouin zone due to umklapp processes with transfer of reciprocal surface lattice vectors gll" In Fig. 15(a) the free-electron-like final-state paraboloid is plotted in the reduced zone scheme with the parameters of the copper fcc-lattice. In Fig. 15(b) the full-relativistic complex final-state band structure is shown as calculated with its real and imaginary parts. Real parts are plotted only when the imaginary part is not too large (llm{k}l < 30% of the distance FL). In this three-dimensional solid the real part of the complex band structure is very similar to the free-electron-like final state as well. The lifetime of the final state was chosen energy dependent as discussed in Section 3.3 according to Ff = 0.065 • (Ef - EF). For the final-state branch associated with g l l - - 0 the imaginary part of k is determined by the lifetime uncertainty and is additionally enhanced in band gaps as discussed in Section 3.5. The other branches transferred into the first Brillouin zone by umklapp with gll ~ 0 have a larger [Im{k}[ due to the smaller group velocity. In Fig. 15(c) the full-relativistic real band structure is plotted for comparison. Within the one-step formalism the photoemission current in angle-resolved photoemission has the following form: l(kll , El) oc Im{< kl[ , Efl G + AG-~AtG2Ikll , Ef)}.

(27)

The electron propagator and the hole propagator are denoted by the Green's functions G~ and G +. The coupling of the electromagnetic field is given by A - - e / ( m c ) .Ao "p, where A0 denotes a spatially constant vector potential [51]. Temperature effects can be accounted for in the multiple-scattering process by assuming statistically distributed displacements of the scattering centers [25,52]. The vibrations on different sites are assumed to be uncorrelated, and a thermal average can be taken site by site. In this way the scattering matrix for each site is modified by the thermal averaging, which results in temperature-dependent phase shifts [52]. In other words, the scattering matrix is multiplied with the Debye-Waller factor.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

178

~

30

V '

"'"'""

20 LS

10

1

L

r

L

L

F Re~k t

L-0.3 0.0 0.3 Irn~kl [A -1]

L

r

L

Fig. 15. (a) Three-dimensional free-electron-like final-state band structure in the reduced zone scheme of the fcc-lattice. The bands are plotted along the FL-direction with parameters adjusted for copper. (b) Full-relativistic final-state band structure of copper in the FL-direction based on the KKR formalism with complex II0. The imaginary part was taken energy dependent as Im{V0} = 0.065. (Ef - EF). The real part of k was plotted only when IIm{k}l < 0.45A -l. (c) Full-relativistic final-state bands calculated with real II0.

Difficulties arise when the temperature dependence of the matrix elements should be included. Due to the non-k-conserving electron-phonon interactions a three-dimensional integration over the Brillouin zone must be involved. This would need too much computer time to realize a one-step calculation. The approach of Larsson and Pendry [52] was not very successful, it results only in a decrease of intensity with increasing temperature for all energies in a spectrum. This, however, is in contrast to the experimental observations [52,70--77] as will be discussed further below. These one-step theories are so far all based on Schrrdinger's equation, i.e. relativistic corrections are neglected. High-resolution ARUPS spectra have demonstrated that even on materials with low atomic number like copper effects due to spin-orbit interaction are present in the spectra [53,54]. A fullrelativistic reformulation of the one-step formalism is given in [55-57]. The computer code (realized by Braun et al. [59]) was made available to the author. The results produced by programs using the onestep formalism not only include the effects of lifetimes and electron final-state, but also effects due to bulk and surface umklapp processes. Moreover, the matrix elements of the transitions are calculated explicitly. These could not be calculated of course in the simple k±-integration (Section 3.2). It is therefore interesting to compare calculated spectra to the experiment with special attention to the peak intensities. Line shapes should be not very different from those calculated by k j-integration: in Section 3.3 we had determined the energy dependence of Fi and Ff, which is now taken with the same numbers as input for the one-step calculation. Results for Cu(1 1 1) in the FLUX mirror plane are

R. Matzdorf/ Surface Science Reports 30 (1998) 153-206 Cu(111) h~ = 21.2eV T = lOOK FLUX 0 = 0°

one-step calculation

L

@=5

°:I' L

°

|

(9 = 15 °

(9 = 2 0 °

_/ L @ = 30°

I

-6

179

Ij

(9 = 4 0 °

1

f

I

-4

-2

0

-6

-4

-2

0

-6

-4

-2

0

Initial s t a t e energy l e V I

Fig, 16. EDCs calculated using the full-relativistic one-step photoemission computer code supplied by Braun et al. The spectra, calculated for Cu(1 1 1) with different electron emission angle/9 in the FLUX-mirror plane, can be compared directly to the experimental data in Fig. 8. Lifetime parameters were taken as determined in Section 3.3.

reproduced in Fig. 16. The spectra may be directly compared with the experiments shown in Fig. 8. The calculated spectrum at 0 = 0 ° reproduces the shape of the experimental spectrum quite good. The direct bulk-emission peaks are in good agreement to the experiment with respect to both line shape and line intensity. The density-of-states transition (label D in Fig. 8) is found neither in the kz-integrated spectrum nor in the one-step calculation. The secondary electron background is of course not included in the one-step calculation. Going away from normal emission relative peak intensities become more and more different to the experiment. It seems the line width is reproduced quite good, although a quantitative analysis is impossible due to the strongly overlapping peaks. Only the 0 = 50 ° spectrum is again quite similar in shape to the experimental one. All in all, we must conclude that for copper, which is traditionally taken as a test substance [58], the matrix elements are not reproduced satisfactorily. There are several reasons, which can be considered for explanation. Firstly, polarization effects due to Fresnel equations are not included in the calculations up to now. Secondly, matrix elements are very sensitive to the wave functions and its gradients; perhaps wave functions in this formalism are not as good as we hope. Thirdly, temperature effects are accounted for only partly, and inelastic scattering effects may be relevant even at low temperature. Anyway, lifetime effects are accounted for adequately, and can therefore not be responsible.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

180

t-

£

Cu(111)

thick line:

h6J = 2 1 . 2 e V

T = 125K

@ = 0°

thin line:

FLUX 0 = 50 °

t

I

T = 703K

experiment

J

$2

t-

Jk e = 0°

thick line:

one-step

T = lOOK

calculation

rLUX

~

thin line: T = 700K

c-

-6

-4

-2

Initial state energy [ e V ]

0

-6

-4

-2

0

Initial state energy [ e V ]

Fig. 17. Upper panels: Experimental E D C s taken from Cu(1 1 1) at two different sample temperatures. Experimental resolution parameters were A E = 30 meV and A 0 = -t-0.7 °. Intensity at both temperatures is plotted as measured and can be compared directly. Lower panel: One-step calculation of the same EDCs for T = 100 K and T -- 700 K, respectively. Lifetime parameters were taken as described in Section 3.3.

Let us now focus our attention on temperature effects. On Cu(1 1 1) EDCs were measured at different sample temperatures between 100 and 700K. In Fig. 17 some of these spectra are reproduced. It is visible clearly that peak amplitudes decrease with increasing temperature and structures in the spectra are "melting away" at high temperature. We observe both loss of intensity in the center of bulkemission peaks and increasing intensity nearby or between the peaks. The lower spectra are calculated within the full-relativistic one-step formalism at the corresponding temperature. The decrease in amplitude is obviously underestimated. This can be explained by the fact that only the temperature dependence of the multiple-scattering process is included in the model. The temperature dependence of the matrix element, due to inelastic electron-phonon scattering, which integrates somewhat over the Brillouin zone, is not accounted for. It is this effect that enhances the loss of peak amplitude and produces increasing intensity in some other energy ranges. It will be the topic of Section 3.7. In some photoemission spectra one can observe different temperature effects, which are dominated by temperature-dependent multiple scattering [60--62]. These are described well by the one-step calculations [61,62]. In spectra recorded on Au(1 1 1) at different polar angles 8 in the FLUX mirror plane with hw = 21.2eV we observed a bulk direct emission peak that changes intensity drastically with O. It has a clearly maximum intensity at 0 = 20 ° (see Fig. 18(b)) and was therefore called an intensity resonance. The experimental spectrum with the corresponding peak labeled A is displayed in

R. Matzdorf/Surface Science Reports 30 (1998) 153-206 I

Au(11'1) ' (~A' ha~ = 21.2eV FLUX, 0=20 ° .

"-2.

II ~I [ ~

•

' a) thick line T = ,.3OK thi: line

I

181 I

I

t.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4-

0.2

0.2

I

c9 = 10( °

I

- ~

O= 5°

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

A

One-step calculation

d)

thick line T = OK thin line

1.0

1.0'

0.8

0.8

0.6

0.6

0.4

0.44

0.2

0.2~

o

f)

c

I

-6

-5

-4

-3

Initial state energy [eV]

I

I

1

0 10 2O 30 Emission angle [o]

G----0°

I

I

2OO 4OO 6OO Temperature [K]

Fig. 18. (a) Experimental EDCs taken from Au(1 1 1) at two different sample temperatures at polar angle 0 along the FLUXmirror plane of the bulk Brillouin zone. (b) Intensity variation with polar angle 0 of peak labeled A in (a). (c) Temperature dependence of peak A at different emission angles 0. (d-f) The same calculated within the framework of the full-relativistic one-step model.

Fig. 18(a) at two different sample temperatures. Peak A has a much stronger temperature dependence of its amplitude than the other peaks in the spectrum. The temperature dependence was studied at different electron emission angles i.e. at different points on the "resonance curve". As a result we find in Fig. 18(c) that the loss in intensity is most drastic at 0 = 20 ° in the center of the resonance. The enhanced peak intensity in the intensity resonance is produced most likely by an interference of the outgoing electron wave with waves reflected in the crystal. The reflectance of the bulk crystal is enhanced at energies in band gaps (see Fig. 14), more generally at energies where crossings of real parts Re{k} for two bands occur [25]. If we imagine the electron final state as several plane waves starting in different directions (corresponding to different reciprocal lattice vectors), we have only one wave moving directly into vacuum. The other waves can be reflected in the crystal or otherwise they are absorbed. Reflected plane waves interfere with the direct one and photoemission intensity can be enhanced drastically in this way. This depends of course on the phase shifts. Due to the temperature dependence of reflectance the enhanced photoemission intensity becomes strongly temperature dependent, too. This effect is, in fact, included in the one-step calculations by temperature-dependent multiple scattering. In Fig. 18(d) the calculated spectra are displayed. The intensity resonance (panel e) is reproduced very well by theory as well as the temperature dependence at different emission angles (panel f). Intensity resonances were also observed on Au(100) as function of emission angle [60]. Furthermore, in Au(1 1 1) and Ag(1 1 1) normal emission spectra intensity resonances are observed

182

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

as a function of photon energy [62]. In these cases the band crossings, which are responsible for reflectance have been identified [63-65] in the real band structure. In conclusion, intensity resonances can occur at special points of the electron final-state band structure. They can be identified by their strong temperature dependence of the peak intensity.

3.7. Electron-phonon scattering In our discussion of the photoemission process we have considered up to now only transitions with transfer of reciprocal lattice vectors or, in other words, with k-conservation in the reduced zone scheme. These are called direct transitions. The k-vector is smeared out only by k±-uncertainty, and photoemission is described in a solid with atoms fixed at the lattice points. In a real solid lattice vibrations are present and these can be additionally activated by the momentum recoil of the emitted electron. That process can be efficiently described by absorption or emission of phonons. In the inelastic scattering process energy Eph and wave vector q of the phonon is transferred. In these "indirect" transitions photoemission intensity may be scattered out of the whole Brillouin zone into the final state that is observed in the experiment. The probability of these scattering processes is determined by the occupation of phonon states. In the same way photoemission intensity can be scattered out of the direct transitions. Several approaches to an incorporation of these phonon-assisted transitions were made in the past [49,52,66-70] and were compared to the experiment [54,67,71-76]. Summarizing discussions can be found in [70,77]. Scattering out of the final state by phonons generally results in a decrease of peak amplitudes with increasing temperature. The decrease can be described by the DebyeWaller factor [25,49] 3h2 [z~xkl2

I = I0"

e -2w

with

T2

xdx exp(x) -- 1 4-

2W-- MkBOD

.

(28)

0

M denotes the mass of an atom, kB the Boltzmann factor and ~)D the bulk Debye temperature; Ak is the difference in wave vector, which here is equal to a reciprocal lattice vector of the bulk. In the hightemperature limit 2W can be approximated by 3hZIAkl2 T.

2W-

(29)

MkBOD

The probability of indirect transitions was derived first by Shevchik [49] with the assumption that the charge density of the initial state follows the motion of the atomic cores. He ends up with a formula for the structure factor that includes direct and indirect transitions. With the assumption of an isotropic phonon dispersion and after carrying out the summation over different phonon polarizations, we get the photoemission intensity I cx e -2wldir q- e -2W Z

I(kf -- ki)12 (a~)lind

(30)

q

with the intensity of direct transitions Idir as given in Eq. (12) and the intensity of indirect transitions lind O( Z I"g'/f 12 " 6(Ef - Ei -- hw ± Eph)" 6(If -- ki - G ± q) i, f,G

(31)

183

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

for emission and absorption of a phonon, respectively. The mean amplitude of a phonon mode with wave vector q is given in the high-temperature limit by (a 2) = kaT/(cZq2M), where c is the speed of sound. To account additionally for lifetime effects in this formula both Idi r and lind can be rewritten with the spectral functions of initial and final state as discussed in Section 3.2. As we see, the intensity of direct transitions is reduced by the Debye-Waller factor and additional intensity arises from the indirect transitions. Indirect intensity rises linearly with temperature initially and then decreases as the Debye-Waller factor decreases and multi-phonon processes become relevant. Since the phonons with the smallest wave vectors have the largest amplitudes the transitions involving the smallest deviation from k-conservation are the most probable ones [49]. In other words, indirect transitions are not simply density-of-states-like. This fact is important for the shape of spectra at elevated temperature. Peak amplitudes decrease by loss of direct transitions, and the increase of indirect transitions results in a broadening of peaks. Of course, this broadening depends on the band dispersion in the three-dimensional region around the k-point which corresponds to the direct transition. An incorporation of indirect transitions was realized numerically in the computer code that has been used previously (Section 3.3) for kz-integration. The integration over k± and q is done quickly by a Monte-Carlo integration, where random numbers are generated with the correct distribution of phonon occupation numbers. The results show qualitatively the loss in intensity and the "melting away" of structures in the spectra. A quantitative comparison to experiment is complicated by the fact that the matrix element is taken k-independent within one transition. Due to the integration over a large volume in the Brillouin zone, this assumption is not justified in general. Nevertheless, the spectral distribution of both contributions Idir and lind is given in Fig. 19 for two spectra calculated for a Cu(1 1 1) surface. The spectra including only direct transitions are well known from the previous sections. For the calculation of the indirect transitions we have separated out the Tdependence in Eq. (30) and have performed the summation over q. Then the shape of the function

IMn(kf, ki)

I

.,4h(Ef -- Ei - hoJ "4- Eph ) • / ~ ( k f z - ki± - G ± -}- q ± )

I

I

I

Cu(111)

rLUX

h~

(9 =

=

212eV

dkf~

(32)

I

FLUX

f

/

(9 = 50 °

0°

C

B

I

-6

4 , ...... , . , -4 -2 Initial s t a t e energy [eV]

I

0

-'6

-4 -2 Initial state energy [eV]

0

Fig. 19. Solidline: Energy distribution curves for direct transitions on Cu(1 1 1) calculatedby k~-integration. Dashed line: Indirect transitions calculated in a way such that every electron has emitted or absorbed one phonon.

184

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

describes the contribution of indirect transitions. Fig. 19 shows results calculated with the assumption that every transition is indirect with either emission or absorption of one phonon. For this calculation the same matrix elements were used as for the calculation of the direct transitions. The resulting spectral shape can be compared to the experiments at elevated temperature as given in Fig. 17 (upper panel). The normal-emission spectrum measured at about 700K seems to be described well by exclusively indirect transitions. This is somewhat surprising, since only about 14% indirect transitions are predicted by theory according to Eq. (30). If indirect transitions were really so dominant, multi-phonon processes should be relevant too. On the other hand, we find a peak in the spectrum measured at 0 - - 5 0 ° that has a considerably smaller temperature dependence and is obviously dominated by direct transition (label B in Fig. 19). Its corresponding initial state is sp-like and the wave function is delocalized. Our assumption of a charge density that follows the atomic position is probably worse in this case than it is for the other transitions which correspond to more localized d-like initial states. The latter fact results in a stronger electron-phonon coupling. In conclusion, transitions out of d-like initial states are dominated by one or multi-phonon assisted transitions, more than those out of sp-like states. In the theoretical treatment the effect of finite T is underestimated, in particular for the d-bands. A more detailed theory including multi-phonon processes and k-dependent matrix elements, will be very complex in its practical use. Therefore, for any analysis of experimental data with respect to line shapes and intensities of bulk-emission peaks it is very important to work at sample temperatures as low as possible.

3.8.

Matrix elements

The intensity of direct transitions in angle-resolved photoemission spectra is an interesting quantity since the matrix element Mfi is very sensitive to the wave functions of both the initial and final state. Wave functions in band-structure calculations are expanded as different series of basis functions and in most cases their quality is tested by the quality of energy eigenvalues. A more sensitive test is to compare the transition momentum matrix element Pfi = (flpli) to experiment. Its relation to the photoemission matrix element is given in Eq. (11). An experimental determination of Pfi can be realized even for direct-transition peaks in the d-bands, which are strongly overlapping in many cases. It was demonstrated in Section 3.3 that relative matrix elements can be extracted from the spectra using the method of k±-integration. The spectra must be recorded at a sample temperature as low as possible to suppress contributions of indirect transitions. For a measurement of direction and absolute value of IPnI the matrix element must be analyzed in several spectra recorded with different directions of A. This is realized by polarized radiation incident in different directions. The vector potential At inside the crystal, which is relevant for photoemission, can be calculated from the vector Ai of the incident light by the macroscopic Fresnel equations [78-81]. Note, however, that this approach assumes divA = 0 in Eq. (7). In a system with z in surface normal direction and the yz-plane in the plane of incidence, the components of A at the surface are given by [79] atz ----

2 c o s ~3i sin ~'~i COS ~/3i + ¢ C --" sin 2 ~)i

IAil,

aty =

2V/e - sin 2 ~)i COS ~)i £ cos ~i + V/~ - sin 2 ff)i

IAil

(33)

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

185

for p-polarized light and Atx =

2 cos ~3i

COS ~3i ÷ V/f~ -- sin 2 ~i

Iail

(34)

for s-polarized light. ~i denotes the incidence angle of light with respect to the surface normal and e is the complex dielectric constant e = el + ic2. Experiments on Cu(1 1 0) and Ag(1 1 0) analyzed from this point of view are published in [23,80]. In [23] the direction of Pn has been determined for an sp-like initial state exited by h~ = 16.85 eV on Ag(1 1 0). The measured direction of Pn contradicts the prediction of bulk theory using the three-step model and the non-relativistic interpolation scheme [81-83]. We conclude that a systematic experimental investigation of matrix elements remains a problem for future work.

3.9. Surface emission In Section 3.1 we have neglected the photoemission intensity due to the term with divA in the interaction Hamiltonian. Inserted in Fermi's golden rule one gets the surface emission transition rate

Wfisurf

iM~Urfl2pf

(35)

with the matrix element

IM~urfl2 __ e2hTr 12. 2m2c2 l( fl divAli )

(36)

Classically there is a discontinuity in the dielectric function e at the surface, which, upon differentiation, leads to a term proportional to 6(z - z0) for OAz/Oz,where z0 is the position of the surface [84]. The 6function is indeed broadened and otherwise modified if the correct quantum mechanical response function of the system is taken into account [20,21]. With the simple approximation of divA by a 6function we can rewrite the matrix element [M~,Urf[2 ~ f ¢ ~ 6 ( z - z0)¢i d3r

= f *;(zo)*i(zo)d2r.

(37)

In consequence, the surface emission matrix element vanishes if kfl I ~ kill, so conservation of momentum parallel to the surface is fulfilled for both direct transitions and surface emission. Recently surface emission intensity has been studied experimentally on copper and silver surfaces in normal emission geometry [22,84-86]. Extra emission intensity has been found in the asymmetric tail of a direct-transition peak derived from the nearly free-electron-like Ag sp-states not far from the Fermi edge [22] (see Fig. 20). In this geometry with kll = 0 the asymmetric line shape of the direct-transition peak can be interpreted within a one-dimensional model due to interference of direct-transition and surface-emission intensity [84]. In one dimension the matrix element reduces to

IM~ urf 12 : C//3~ (z0)ff3i (z0)

(38)

with the complex constant C which is treated as a parameter to fit the experimental data. The wave functions are obtained by a two-band fit to the Ag sp-band dispersion using the pseudopotential method. The initial state is a Bloch wave function whereas the final state is a gradually attenuated Bloch wave

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

186

Ag(l 11) Normal Emission (hv = 8 eV) 1

•

~

I

3

I

Surface State-->

• Expt. ---Mo~el

l t

t

2 l Binding Energy (eV)

0

Fig. 20. Energy distribution curve taken from Ag(1 1 1) in normal emission with p-polarized light hw = 8 eV. The curve is a model fit to the experimental spectrum (circles). The dashed horizontal line indicates the zero-intensity level. The dotted curve shows the inelastic background (figure taken from [84]).

function modeled by an exponentially decaying envelope function. This decay is equivalent to the k±uncertainty we have used in the previous sections. The lifetime of the initial state and the instrumental resolution are included in the model by convolution with an adequate function. Thus in this model direct-transition intensity within the one-step formalism as well as surface-emission intensity are calculated simultaneously. Due to interference of the outgoing electron waves I cx IM~fiirect -'1-M~urfl2,

(39)

the shape of the observed emission peak is modified considerably. In Fig. 20 a perfect agreement of experimental and modeled data is demonstrated. Similar good agreement is found in experiments on Ag(1 00) [85]. A further test for surface emission from Cu(1 00) is reported in [86]. The asymmetry of an sp-band "direct emission" peak is suppressed by deposition of a few monolayers of Ag on the surface. In the resulting copper-silver interface the discontinuity of e and A, respectively, is reduced considerably in comparison to the clean copper surface. This results in a drastic reduction of surface emission.

4. 4.1.

Line shape of surface states Historical outline

In 1932 Tamm [87,88] predicted electronic states which are confined between the surface potential barrier and a totally reflecting periodic potential. The wave function of such a surface state is localized at the surface and decays exponentially in both directions within the one-dimensional model. Shockley [89] extended the theory to a more general potential and to the three-dimensional case. He pointed out that in this case the surface state exhibits a dispersion parallel to the surface. But it needed 35 years until the first experimental evidence for the existence of a surface state on a metal surface was obtained

R. Matzdorf/Surface Science Reports 30 (1998) 153-206 Io01] _

X

~ [110} X

_

~, ~: I ~.[111] ~' _

y I

S

S

M

~

Fig. 21.

,.

=

R'

W

I W

187

K W

[d~

.W

X

I

Bulk and surface Brillouin zones of the three low-index surfaces of the fcc-lattice with symmetry points indicated.

[90,91]. In the latter two decades research in electronic structure of surfaces has become a rapidly developing field. Several previous reviews on experimental investigations with angle-resolved photoelectron spectroscopy [5,92-100] and on surface electronic structure theory [101-109] are found in the literature. In this section investigations of surface states on the noble-metal surfaces are reported briefly, since they played an important role in the context of line shape and line width studies during the development of angle-resolved photoemission spectroscopy. In 1970 Forstmann and coworkers [110-112] predicted surface states on d-band metals in different kinds of band gaps, and five years later Gartland and Slagsvold [113] discovered a surface state on Cu(1 1 1) experimentally. It was a "Shockley"-type surface state in the sp-band gap around the L-point of the bulk Brillouin zone. Calculations of surface state bands were published by several authors [ 115122]. They also predicted surface states at other symmetry points of the surface Brillouin zone (SBZ) as well. For a definition of the SBZ see Fig. 21. Meanwhile surface states were experimentally detected and investigated on all noble-metal (1 1 1) surfaces at the F-point of the SBZ [114,123-127]. The dispersion with kll of these states is parabolic around I" [113,128,130]. First measurements on the temperature dependence of the surface state emission peaks were performed [131,132] and a temperature-dependent energy shift was detected on Cu(1 1 1) [132] and explained in [52]. Surface state bands of the Shockley-type located in band gaps at the Y-point of the SBZ of Cu(1 1 0), Ag(1 1 0) and Au(1 1 0) were observed experimentally [128,129,133] as well as surface states at the top of the bulk d-bands of Cu(1 00), Cu(1 1 1) [134--136,159], Ag(1 00) [136--140], Au(1 00) and Au(1 1 1) [141]. These latter d-like states are of " T a m m " - t y p e since they are located in a non-hybridizational gap of the projected bulk band structure [134]. Their existence requires that the surface perturbation of the oneelectron potential is sufficiently strong [135]. The dispersion E(kll ) of the Tamm states is parabolic around the 1Vl-points of the (1 00) and (1 1 1) surfaces, respectively, with a negative effective mass. While experimental energy and angular resolution was improved more and more, attention was focussed on the line shape of the surface states. High-resolution ARUPS studies were carded out by Kevan and coworkers [6,142-144] on the Shockley and Tamm surface states of copper. Their discussion of broadening mechanisms and non-lifetime effects will be continued in the following

188

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

sections. Only with very much improved energy and momentum resolution Kevan [145] was able to detect a further Shockley state located near the Fermi energy at the ,'~-point of the SBZ of Cu(1 0 0). Experimental binding energy and effective mass were found to be in excellent agreement with selfconsistent slab calculations [146]. Furthermore, two new surface states located in s--d hybridizational gaps in the projected band structure of copper are observed by Kevan et al. The first is centered near the X-point of the SBZ of Cu(1 0 0), the other one is located near the Y-point of Cu(1 1 0) [147]. They are qualitatively similar to the Shockley state observed with Ei --- - 0 . 4 e V at r" on Cu(1 1 1) [126]. However, these states with binding energies > 4 eV have a considerably larger line width as compared to the other ones mentioned before [147]. This is most probably the fingerprint of a comparatively small hole lifetime. With the technique of inverse photoemission unoccupied surface states were observed with binding energies between EF and the vacuum energy Ev. In [148-150] Shockley-type surface states and imagepotential states on the noble-metal surfaces are reviewed. In some cases these states are observed to continue the dispersion of the occupied Shockley-type surface states [151,153]. A high-resolution technique for probing these unoccupied states is two-photon photoemission [152]. Some results concerning the lifetime of image-potential states will be discussed in Section 4.7. The occurrence of image-potential as well as Shockley-type surface states was analyzed by Smith [154] with use of a combination of elementary multiple-reflection theory and elementary nearly freeelectron theory. Within this "phase model" [155-158] the binding energies of the surface states in nearly free-electron gaps can be calculated at the symmetry points of the SBZ using bulk band-gap parameters and position of the surface barrier as input. A direct measure of the scattering cross section off trace potassium surface impurities of the electrons bound in the F surface state on Cu(1 1 1) is described in [160]. Final-state scattering as a source of momentum broadening has been ruled out [ 160]. The binding energy of the surface state on a potassium covered surface and its dependence on substrate work function was semiquantitatively reproduced by the phase model [ 161 ]. Most of the measurements reported above are carried out at or above room temperature. Only recently, several investigations of the temperature dependence of surface state properties were published. Changes in intensity and line width of surface state emission as well as energy shifts of the surface state bands on the noble-metal surfaces were investigated in a temperature range of 40-700 K. Detailed results for the Tamm surface state on Cu(1 00) and a corresponding state on Cu3Au(1 00) were given in [54,162]. In [163,164] high-resolution data of the Shockley-state on Cu(1 1 1), Ag(1 1 1) and Au(1 1 1) were presented. In particular the line shape of the surface state emission peaks was discussed in detail with respect to the influence of energy and angular resolution, sample temperature and the proximity of the Fermi edge. Using the phase model, the linear shift of binding energy with temperature was explained. It is quantitatively related to the temperature-dependent band-gap parameters. In particular on Ag(1 1 1) the surface Fermi contour becomes temperature dependent due to the energetic proximity of the surface state to the Fermi edge. An influence of the temperaturedependent occupation of the surface state on the surface geometry is discussed in [166]. Recently, the Shockley surface state on Au(l 1 1) has been found to be a doublet [167]. As origin of the splitting LaShell et al. [167] propose spin-orbit coupling, resulting in two spin-split surface state bands with the spins aligned in the plane of the surface perpendicular to the electronic momentum. In a high-resolution study of the Tamm-state at M on Cu(1 1 1) [168] this state has been resolved for the first time both at 1(,I and 19I' (compare Fig. 21). The line width of this surface state emission peak is

189

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

Ef(ki)-h~~

Cl)

Akfl....~/ / /I//I

El(k±)

~ . . , -

_

&E

t

~

-'--..-.

/

_"_.'-._.'b~__Y_v / ............... "- - ~ , . . / .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

b)

/

/

/.~//

~kft

/

/ iiII

_?' __,'/_,,_'_

/

/ ZXE,

.

E~(k~)-.~ ~

_

iii/i II ~"...~ -

// / //

&E,

6 1 1 / 1 II kl

ki

Fig. 22. Schematic diagram of contributions to the measured full width at half maximum (FWHM) AE for bulk-emission peaks (a) and surface state emission peaks (b). The dispersion Ei(k±) of the initial state is plotted as well as the dispersion Ef(kx) of the final state shifted down in energy by the photon energy. The direct transition is found at the crossing of both bands. Furthermore, the uncertainty in energy of Ei and in k± of Ef are indicated as dashed lines, both given as FWHM of the spectral functions.

measured to be 28 meV at T = 105 K. The surface state emission peak at lVl of Cu(1 00) was measured with a width (FWHM) of 2 8 m e V [159], and very recently with 23meV at T = 100K [165]. The intrinsic width of the Shockley surface state on Ag(1 1 1) has been determined to be < 20meV at T = 56 K [164]. An extrapolation to T ---* 0 of the/Vl-state on Cu(1 00) suggests a FWHM of < 13 meV [169]. These are to our knowledge the smallest photoemission line widths observed from occupied states on solids.

4.2. Relevancefor lifetime investigations The preceding section shows that photoemission lines originating from surface states are often much narrower than those due to bulk band transitions. Therefore they were always preferred candidates for line shape studies, and in what follows we will discuss some reasons for that. In Section 3 we had already discussed in great detail the influence of initial-state and final-state lifetime on the line shape of bulk-emission peaks. The k±-uncertainty of the final-state wave function gives rise to a broadening of the emission peak. For normal emission geometry this effect is visualized in Fig. 22(a). Due to the fact that surface states do not exhibit any dispersion with k± the line width of surface state emission peaks is not influenced by k±-uncertainty (Fig. 22(b)). The intrinsic line width represents exclusively the lifetime broadening of the initial state. In addition, however, the measured line width of surface states depends also of course on experimental energy and angular resolutions as well as on the concentration of surface imperfections, and eventually other sources of broadening.

4.3. Modification of line shapes by instrumental resolution To investigate line shapes near EF the Shockley surface state on Cu(1 1 1) was studied for several times in the past [6,142]. It has a binding energy of --~ 400 meV at F and disperses upwards crossing EF

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

190

EF=O

98meV

Cu(IT--'--~I )r

o)

-100

Contribution

E

Hel 0 E 0

to linewidth

/..i.. ~

-200 59meV -~

o ffl -300 o c

12meV ~

-400

2meV

-0.2

-0.1 Porclllel

I

I

1-I

0.0 0.1 0.2 momentum kll [A-l] I

o:oo 1.5

110meV

I

i

i

I

i

b)

Jl 0=3° T=115K H A Cu(111)

/" k

/I \ IA \

"~

T-- 300K Hel

(~ 1.0

I.~e:~o / / , t

0.5

\

IE ^~.

-0.6

-0.4 Initial store

-0.2 EF=O energy [eV]

-0.6

-0.4 Initiol store

-0.2

EF=O

energy [eV]

Fig. 23. (a) Dispersion E(kll) of the Shockley surface state on Cu(1 1 1), experimental data (filled circles) and parabolic fit (solid line). Contributions to the line width of the photoemission peak measured at different angular resolutions (solid vertical lines) in normal emission are indicated on the right schematically. Dotted lines indicate the kll range which contributes to measurements in off-normal direction with A0 -----+0.4 °. (b) EDCs of the surface state measured at different electron emission angles. Experimental data are plotted as circles, boxes and triangles, respectively. The solid line is the result of a twodimensional kit-integration, see text for details. (c) EDCs of the surface state measured with different angular resolutions. Intensities are normalized to same peak amplitudes. The vertical dashed line indicates the initial-state energy measured with best resolution.

at kll = 0.21/~-1, s e e Fig. 23(a). For this system it should therefore be possible in principle to observe Fh as Ei ~ EF. In fact, the Landau theory of Fermi liquids predicts Fh ~ 0 when Ei approaches EF = 0. A simple argumentation is as follows: If the hole lifetime is governed by decay via Auger processes, Fh should be proportional to the number of occupied states between the hole-state energy Ei and EF. Phase-space considerations [171] then predict Fh cx (Ei - EF) 2. However, numerical estimates indicate that within 0.1 eV to EF, the lifetime width is less than 1 meV. This has been discussed recently in great detail with respect to the F state on Cu(1 1 1) by Smith [8]. These data indicate that Fh may be

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

191

easily hidden below larger contributions to F by instrumental effects. It is therefore very important to discuss and understand the influence of the resolution parameters. Excluding phonon and defect scattering, the intrinsic line shape of a surface state is given by the spectral function of the hole state ,Ah. It is broadened due to the experimental energy resolution. The resulting line shape is given as a convolution with the instrumental resolution function Dr(E). It includes the energy resolution of the analyzer as well as the line width of the UV-radiation. The finite angular acceptance cone of the detector is correlated with a finite area in the SBZ where electrons are collected from. This may be expressed by an instrumental acceptance function in wave vector Dk(k N). The measured intensity distribution is then given by

I(gvan,kt] ) oc ]~lfi(kll,tiw)[2 f 79k(klL- kll ) • .Ah * (mt(gkin q - ~ -

h a ; - gi(kll ))dkl] ,

(40)

again with the assumption that the matrix element hS4n(kH, ha;) is constant within the kll-area from which electrons are accepted. The influence of angular resolution on line shape is therefore depending on the dispersion of the surface state band with kll. In Fig. 23(b) measured energy distribution curves of the Shockley surface state on Cu(1 1 1) are shown for different electron emission angles 0. We observe an increase in line width when the peak is approaching EF, in clear contrast to the prediction Fh ~ 0 of the theory of Fermi liquids. The energy resolution of about AE ~ 25 meV has no significant influence on the line shape of the measured peaks with a FWHM of 57 meV and more. To check for the influence of angular resolution, we performed a two-dimensional kll-integration according to Eq. (40). For this purpose we have assumed that the intrinsic line shape .Ah is a Lorentzian with a constant width Fh = 57 meV, independent of Ei. With an angular resolution of dz0.4 ° the calculated line shape (solid line) fits the experimental data very good. In the 0 = 5 ° spectrum the matrix element has also been reduced by 12% (dotted line) with respect to the 0 = 0 ° spectrum in order to get the best fit. Furthermore, a linear background and the Fermi distribution function were accounted for in the calculation. It is clearly visible that the angular resolution can explain the observed effect of peak broadening completely. In a recent experiment McDougall et al. [170] have measured a line width at F of 33 meV at T = 30 K. In the off normal spectra the width increases by less than 5 meV. In this experiment the angular resolution in direction of surface state dispersion was set to +0.06 °. Since a narrowing of the surface state while approaching EF is not observed even in this very high-resolution experiment, additional broadening mechanisms may be relevant. We will come back to this in the following sections. In first order approximation broadening due to the finite angular resolution vanishes if the dispersion XTEi(kll) = 0. This is given for the Shockley surface state in the center of the SBZ. In Fig. 23(c) three spectra of the surface state are plotted, measured with different angular resolutions. A significant broadening due to second order terms becomes visible at angular resolutions I/x01 ___ ±1 o Besides an asymmetric broadening due to the parabolic dispersion an energetic shift of the peak maximum is observed additionally. All in all we must conclude that an excellent angular resolution is necessary if lifetime information is to be deduced from measured line widths of these rapidly dispersing surface states.

4.4. Electron-phonon interaction In order to deduce hole lifetimes from peak widths of surface state transitions reliably, also other broadening mechanisms must be taken into consideration. First of all there will be contributions from

R. Matzdorf/Surface ScienceReports30 (1998)153-206

192

indirect transitions. In these transitions (emission or absorption of a phonon) the wave vector q and energy Eph of a phonon are transferred. The wave vector component q± perpendicular to the surface does not change anything in the measured final-state energy. However, due to transfer of qll electrons with different initial-state energies are scattered into the cone that is observed by the detector. According to Eq. (32) indirect transitions contribute with 1

Iind(Ef, klr) oc ~ ~

I.(,/n(kll , ha~)l 2 • . A h ( E f -- E i - ~a3 -Jr Eph),

(41)

where Ei is given by Ei(kll + qfl). The intensity due to indirect transitions rises with increasing temperature. The discussion about the temperature dependence of direct and indirect bulk transitions given in Section 3.7 is in general valid for surface states, too. In the surface things can be modified somewhat due to surface phonons and enhanced surface vibrations expressed by a surface Debye temperature smaller than that of the bulk. Temperature-dependent experiments on the Shockley surface state on Cu(1 1 1) are shown in Fig. 24(a), which reproduces a set of spectra measured at different sample temperature. The energetic shift of the surface state band is understood well in terms of the T-dependence of the bulk band gap that

A II

13)

b)

Cu (111) r

I l

I I Shockley surface state l 1

t I

ha) = 21.2eV e=o °

T = 128 ..........

-0.6 -0.4 -0.2 Initial state energy [eV]

-0.8

i

1.0

0.8

"~

¢)

~.T= 675K -0.8

-0.6 -0.4 -0.2 Initial state energy [eV]

I

EF=0

EF=O

0.6

0

dr'.

i 200

I

4OO Temperature [K]

l

I

6OO

Fig. 24. (a) EDCs of the Shockley surface state on Cu(l 1 1) measured at different sample temperatures. To guide the eye a symmetric Lorentzian (solid line) is plotted through the experimental data (dots). (b) Shape of direct (solid line) and indirect transitions (dotted line). See text for details. (c) Dots: Intensity (peak area) versus temperature taken from the data in (a). Solid line: Temperature dependence of direct emission intensity predicted by the Debye-Waller factor calculated with the bulk Debye temperature. Dashed line: Debye-Waller factor with an adjusted ~ D = 110 K.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

193

supports the F-state [164]. Secondly, we observe a loss in peak intensity. It is due to the decrease with increasing temperature of transitions involving no phonons. Fig. 24(c) gives the measured peak intensity as a function of temperature. Intensity decreases more than predicted by the Debye-Waller factor (Eq. (28)) if calculated with the bulk Debye-temperature of (gD = 343 K (solid line). The Debye temperature ~9D must be reduced by more than a factor of 3 to get a good fit to the experimental data (dashed line), but the used fit-parameter tgD = 110 K is much smaller than typical surface Debye temperatures determined from LEED experiments [25]. As also observed for the bulk-emission peaks (Section 3.7), the temperature dependence of peak intensities is stronger than predicted by a model with a simple Debye-Waller factor. Thirdly, in the spectra of Fig. 24(a) we see that the peak width increases with increasing temperature. The peak shape is almost Lorentzian-like even at high sample temperatures. Due to the parabolic dispersion of the surface state indirect transitions can contribute to intensity only in the energy range between peak maximum and Fermi energy. According to Eq. (41) we have calculated the shape of the indirect transition intensity. The spectral function of the initial state was assumed to be Lorentzian-like with a FWHM = 57 meV according to the low-temperature measurement. Indirect transitions are plotted in Fig. 24(b) as a dotted line in comparison to the direct emission peak (solid line). Intensity due to indirect transitions is comparatively small in this case since the part of the surface Brillouin zone, where the surface state is occupied, is quite small. Due to their fiat asymmetric shape the indirect transitions cannot be responsible for the broadening of the surface state transition peak at elevated temperature. Rather we suggest that the peak broadening is caused by a temperature-dependent initialstate lifetime. As a second example in Fig. 25 experiments on the Tamm surface state on Cu(1 1 1) are shown. This surface state is much more localized and has d-orbital character. The dispersion (see Fig. 25(b)) of this state around the/Vl-point is parabolic too, but considerably weaker than that of the Shockley-state, compare Fig. 23(b). The effective mass m*/m = - 2 . 8 6 + 0.2 of the Tamm-state is larger by about a factor of 7 in its absolute value compared to the Shockley state (m*/m = 0.41 + 0.02). In the spectra measured at different sample temperature (Fig. 25(a)) again we see that peak intensity is reduced and width is increased with rising temperature. A temperature-dependent background arises from a bulk transition peak at Ei = - 2 . 2 e V . The peak broadening at elevated temperature cannot be due to indirect transitions since the calculation according to Eq.(41) gives only a weak asymmetric feature reproduced in Fig. 25(c). It is in practice impossible to distinguish in this spectrum between indirect transitions of the bulk-emission peak and those of the surface state. We conjecture that the peak broadening is due to a temperature-dependent initial-state lifetime.

4.5.

Lifetime of the hole state

In our discussion of lifetime effects we have up to now assumed the hole lifetime to be only dependent on the initial-state energy. The Landau theory of Fermi liquids predicts a quadratic dependence of inverse hole lifetime on energy/~h =/3(Ei - EF) 2. An analytic approximation for the imaginary part of the self-energy of a quasiparticle in jellium has been given by Quinn [ 171 ]. It can be expressed simplified by [8] I m { E } _~ 1.1 × lO-3rSs/2(Ei- EF) 2,

(42)

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

194

I

I

o)

I

I

I

I

I

i

I

I

I

1.7

1.8

cu (111)

Y

/

g

CoCl11

%

2m "6 E

1.1

1.2

F~

-1.8

T 1.5

~.6

a i

-2.0 -1.9 Initial state energy [eV]

~.3

k,, [A -1] I

-2.0 -1.9 Initial state energy [eV]

I

-1.8

Fig. 25. (a) EDCs of the Tamm surface state at/Vl on Cu(l 1 1) measured at different sample temperatures. We assume a smooth background (dashed line) due to bulk-emission intensity. (b) Measured dispersion Ei (kll) of the Tamm-state around the IVI-point of the SBZ (filled circles) and a parabolic fit (solid line). (c) Shape of direct (solid line) and indirect transitions (dotted line) as calculated using Eq. (41).

where rs is the conventional jellium density parameter and energies are given in eV. Taking this relation as a rough estimate for copper with rs ----2.67 we get a value of fl --- 0.013 eV -] . This fl is larger only by a factor of 2 than the value fl = 0.006 eV -1 which has been determined experimentally for bulk states at Ei < - 2 eV on copper as discussed in Section 3.3. The almost quantitative agreement is surprising in view of the fact that d-holes in Cu are probably no good representatives of quasiparticles in jellium. With respect to the Shockley-state at F on Cu(1 1 1), Eq. (42) predicts Fh < 2 m e V for IEi - EFI < 0.4 eV. This is obviously much less than observed in recent experimental spectra. Eq. (42) has been derived at T --- 0 and accounts for the Auger decay of the photohole. At finite temperature, one must take into account an additional term [170,172]. Fh =/3[(TrkBT) 2 + (Ei

-

-

EF)2],

(43)

which contributes with only less than 1 meV in the relevant temperature range. In addition the hole state can decay via the electron-phonon interaction into a less tightly bound hole plus a phonon [170]. The inverse lifetime of this contribution Fep can be expressed by [170,173] Cdmax

/~ep = 7 r h / t 7

0

a 2 F ( J ) [ 1 - f ( w - a;') + 2 n ( J ) + f ( w + a/)] d~',

(44)

R. Matzdorf/ Surface Science Reports 30 (1998) 153-206 I

30

I

I

i

o)

T=4OOK

~

b)

i

195 I

I

50

.y:-" F/.,'

~, ©

25

300K

L_ o4

2OOK

e~ 10

lOOK

"/-'"

~g 30

~

y,",/,,."

:" ~'/""

20

~

10 I

[ .........

'~'--------;,r-

L_

-400 -300 -200 - 1 O0 Initiol store energy [rneV]

Fig. 26.

~.

40

.

0

E :=0

'""

0

I

200

I

400 Temper(]~ture [K]

I

600

(a) Solid line: electron-phonon contribution to the hole state decay width 2Fep as a function of initial-state energy

calculated in the Debye model with parameters of copper (WD= 27 meV, A = 0.14) for different temperatures. Also shown is the quadratic energy dependence of 2Fh due to Auger-decaycalculated with fl = 0.013 eV-] (dashedline) and fl = 0.006eV-l (dotted line). (b) 2Fh as a function of temperature calculated within the Debye model for Ei = -400meV (solid line) and Ei -- EF = 0 (dotted line). Dashed line: linear approximation in the high-temperature limit according to Eq. (45).

where w is the phonon frequency, o~2F(w) is the Eliashberg coupling function, a n d f ( w ) and n(w) are the Fermi and Bose-Einstein distribution functions. In the high-temperature limit with n(w)~_ (kaT)/(hw) and n(w) >> 1 > f ( w ) Eq. (44) reduces to Fep ~' 27rAkaT

(45)

with the electron-phonon mass enhancement parameter A. This parameter is in principle also temperature dependent according to A = 2 f~0m~"ot2F(w)/wdwat zero temperature and A ~ 0 at very high temperatures, but this temperature dependence is neglected in the following. Theoretical estimates for A of the noble metals are given by Grimvall [174]. With A = 0.14 for copper contributions to the inverse lifetime of several 10 meV are expected in the vicinity of the Fermi energy. In the Debye model the energy and temperature dependence of Fep Can be calculated easily using olZF(w) ---- A(W/WD)2 for < ~D and zero elsewhere [170]. The result is displayed in Fig. 26. From Fig. 26(a) it becomes clear that phonon scattering contributions to the hole lifetime (solid lines) close to EF are larger than the contributions by Auger-decay (dashed and dotted lines), even at zero temperature. A measurement of the Auger contribution to the quasiparticle lifetime and its energy dependence is therefore quite hopeless in this energy range. In contrast, the temperature dependence of Fep can be measured well. Fig. 26(b) shows Fep calculated in the Debye model for two different binding energies and using the approximation of Eq. (45) (dashed line). The linear approximation is quite good for temperatures above T = 100 K and can be used for the interpretation of experiments. Fig. 27 reproduces the measured line widths of the Shockley-state on Cu(1 1 1) as a function of temperature. Data were taken from [170,175]. From both data sets A is determined to be 0.14 ± 0.2 eV. This value fits very well to that given by Grimvall [174],

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

196 100

>Q)

I

100

o)

b)

'

80

~'

80

•"9-

40

0u(111) Tamm surface state

E -r

I

60

It.

'i~

/

40

./

•

0

Cu(111) f Shockley surface state

O

o_

20

0

200

400 600 Temperature [K]

800

20

0

200

400 600 Temperature [K]

800

Fig. 27. (a) Width of the Shockley surface state emission peak on Cu(1 1 1) measured with ArI-radiation as a function of temperature (dots) [175]. Solid line: linear fit to the experimental data. Dashed-dotted line: experimental result of McDougall et al. [170]. Dashed line: estimate of Auger-decay contributions to the lifetime broadening. (b) Width of the Tamm surface state emission peak taken from data displayed in Fig. 25. Error bars account for the uncertainty in subtracting background. Solid line linear fit with A = 0.085.

who reports an average over the Fermi surface of A = 0.15 from completely different experiments. A closer analysis makes this apparent agreement a little less clear as discussed by McDougaU et al. [170] since A varies from 0.085 to 0.23 depending on the position on the Fermi surface [176,177]. In general A is depending on k and Ei. A measurement at a special point in the Brillouin zone can be realized by photoemission experiments provided that a surface state is present there. On the example of the Tamm-state on Cu(1 1 1) we demonstrated that even far away from the Fermi surface (at Ei = - 1 . 9 3 eV) the increase of Fep is linear with T and A can be determined reliably [175]. Fig. 27(b) reproduces the experimental result, showing a slope of A -- 0.085 + 0.01. It is an open question whether the electron-phonon mass-enhancement parameter is the same measured at a surface and a bulk state, respectively. Due to the localization of the surface the electronphonon coupling may be in principle quite different to that in the bulk. To answer this question further experiments are required. Despite the electron-phonon scattering we can extrapolate the peak width to T = 0 K to determine the contributions due to lifetime broadening• In Fig. 27(a) the contribution of Auger-decay is indicated by a dashed line including the temperature dependence of Ei of the Shockley surface state and the temperature dependence of Fh itself. It is quite small at any temperature. At T ----0 K a contribution to the line width of more than 30 meV is still unexplained. It is most probably due to surface imperfections like defects and contaminations. In the measurement of McDougall this contribution is smaller by about 15 meV because the experiment was performed on a small sample area of only about 150~tm in diameter. This has been realized by focussing the UV-light and selection of a "best quality" spot on the sample to obtain the narrowest line at F. At this point of discussion we must be aware of the fact that in the determination of A any contributions by surface imperfections have been assumed to be temperature independent.

R. Matzdorf/Surface Science Reports 30 (1998) 153-206

197

In the case of the Tamm surface state the contribution by Auger-decay estimated from Eq. (43) is larger than the extrapolated peak width at T ---- 0 K, This is an experimental indication that electronelectron coupling, i.e. Auger-decay is also different for surface and bulk states, respectively.

4.6. Defect and impurity scattering The incorporation of surface impurity scattering in the interpretation of angle-resolved photoemission data has been presented first by Tersoff and Kevan [6,142,160]. They discussed the line shape of the Shockley-type surface state on Cu(1 1 1) covered with trace potassium impurities. The electrons in this surface state are scattered by the impurity atoms distributed randomly on the surface in small concentration. Due to these elastic scattering processes the wave vector kll is changed, whereas Iklll is left unchanged. In consequence, all wave vectors with the same Iklll contribute to the distribution of kll of the eigenstate with energy E = li2JkllJ2/2m*. In addition, the impurities endow the surface state electrons with a lateral mean free path l, which results in an uncertainty of the absolute value of kll, given by O'imp 1/1 [160]. This line width contribution to measured photoemission peaks is quite similar to the effects due to experimental angular resolution as discussed before. In first order approximation a resulting broadening on the energy scale can be calculated by [160] "~

~E Fimp = ~

(46)

Oimp.

Fig. 28 displays schematically the dispersion of the Shockley state with an uncertainty in Ikll t indicated by dashed lines. In a measurement at sharp kll an interval of different initial-state energies is measured

EF=O

I

Cu(111) >

-100

E

Shockley surfoce store I =

IOOA

)...,

/

./

-200

S (/1

/

AEimp

"

-300i c

j J'/

/'

-4-00 I

0.0

0.1 0.2 Absolute of parallel momentum Ik,l [ A t ]

Fig. 28. Dispersion E(lkll J) of the Shockley surface state on Cu(1 1 1) (solid line). The uncertainty in IkilJ is indicated with dashed lines for a surface with a lateral mean free path of l = 100 A. The resulting energy broadening A E is indicated on the right-hand side for a measurement at kll = 0.15 A - ].

198

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simultaneously, which results in the energy broadening Z~imp indicated on the right-hand side. An impurity concentration that reduces the mean free path l to 100,~ results in about AEimp = 2Fimp ----56meV at kfl = 0.15 ~ - l . In [160] the line width of the surface state emission peak was measured at kjl = 0.15 A -1 for various potassium coverage in the range of 0-0.02 monolayer (ML). The lateral mean free path is reduced from l > 300 A on the clean surface to l = 50,~ at 0.02 ML potassium coverage. Another interesting observation, also pointed out by Kevan, is that the observed broadening effect is independent of the final-state energy. This rules out final-state scattering as an effective momentumbroadening mechanism [160]. This fact agrees very well with our own experimental result on vicinal surfaces, discussed in Section 3.4: a scattering of the final-state electrons at step edges was not observed for bulk-emission electrons, too. It is indeed a modification of the initial-state wave function that results in the peak broadening due to defect or impurity scattering as measured sensitively by angle-resolved photoemission. Recently, standing waves of the Shockley surface states on the noble-metal (1 1 l) surfaces have been imaged in real space by scanning tunneling microscopy (STM) [178-186]. A scattering of the surface state wave function at point defects and step edges is clearly visible in the STM images, which probe the local density-of-states of the surface states [178,182,183,185]. In [179] an image is given that displays the radial standing wave surrounding an Fe adatom on Cu(1 1 1) due to scattering of the surface state electrons. The most prominent pictures in this context are those of the surface state wave function confined into a quantum corral constructed as a 48-atom Fe ring on Cu(1 1 1) [179]. The surface state wave function on clean copper surfaces is confined on the terraces by reflection at the step edges [178,185]. Due to this confinement the wave vector of the F surface state eigenfunction becomes uncertain, which results in a peak broadening as discussed above. Additionally, the lateral confinement shifts the binding energy of the eigenstates towards EF. Experiments on stepped Cu(1 1 1) surfaces studying the line width of the F surface state photoemission peak are reported in [187]. The stepped surfaces were copper vicinal surfaces with a miscut of 5.5 ° and 8 ° with respect to the I1 1 1] direction. The energetic shift in binding energy of the surface state due to this one-dimensional confinement was found to be 80 meV on a terrace with a width of 15.4 ,~. In addition to the energetic shift an increase in line width is found with increasing miscut. Three different effects could be responsible for this line broadening. Firstly, the uncertainty in wave vector, secondly, a broad distribution of different terrace sizes all contributing with different binding energies, and thirdly, a reduction of the photohole lifetime due to scattering of the hole into bulk states. On the vicinal surfaces the direction of k± is different from the [1 1 l] direction, as discussed in Section 3.4. This has as consequence that the projected band gap is not at all present and the surface state becomes a surface resonance. The broadening due to the first effect has been calculated by Beckmann et al. [188], it results in an asymmetric line shape. In the calculation the line width is increased from 30 meV on the well-ordered surface to 70 meV on a stepped surface with geometrical distribution of terrace widths and a mean terrace width of 20.~. Sfinchez et al. [187] have observed an increase of line width from 150 to 270meV on the stepped surface. These effects are even relevant on well-prepared surfaces used in high-resolution ARUPS experiments. The Cu(1 1 1) crystals are in general oriented not better than 0.25 ° and the lateral mean free path is reduced by impurity point defects. All these broadening effects contribute to the measured line width of the surface state emission peak at any sample temperature. In consequence, the

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preparation of excellent single-crystal surfaces is an indispensable requirement for further lifetime investigations exploiting photoemission from surface states.

4. 7.

Lifetime of image potential states

In addition to the occupied surface states another class of electronic states located at the surface is found on the noble-metal surfaces. In the energy range between EF and the vacuum level Ev unoccupied states exist in the potential well built by the image potential and the band gaps of the projected bulk band structure. In this quantum well a Rydberg like series of image states is found. These states labeled with their quantum number n = 1 , 2 , . . . are very similar to the Shockley-type surface state labeled n = 0 in this context. The unoccupied states are not accessible by angle-resolved pbotoemission, they had been detected by inverse photoemission (IPE) [189-191]. Recently, these states have also been studied by two-photon photoemission (2PPE) [ 152,192,193], which is much better in the experimental resolution parameters. Recent experiments have demonstrated energy resolutions of 30meV [194,195] and an angular resolution of 0.007~, -1 [196,197]. Combined with ultrafast laser pulse techniques, time-resolved 2PPE provides a powerful technique to directly probe the time evolution of image states on the femtosecond time scale [198-200]. In recent experiments lifetimes of T = 10fs have been analyzed with a laser pulse duration of 60fs [199]. In 2PPE measurements an electron from an occupied state below EF is exited by a photon into an unoccupied image state, realized by a laser pulse with photon energy smaller than the work function in order to suppress one-photon photoemission. Photons of the second laser pulse arriving within the lifetime of the exited state lift the electron above the vacuum level where it can be analyzed like in ordinary photoemission. With a delay between both pulses the time-dependent population of the exited state can be measured, Image potential states are expected to have long lifetimes, because they are located mainly in front of the surface, which results in a small overlap with bulk electron wave functions. A particular advantage of 2PPE is that the lifetime 7- measured on the time scale can be compared directly with the lifetime broadening F = ti/7- in energy. As an example we pick out the n -- 1 image potential state on Cu(1 1 1). It is measured with 85 :k 10meV in energy-resolved 2PPE corresponding to a lifetime of 7- = 8 i 1 fs [193]. The lifetime obtained by direct time-resolved measurements is Tl = 18 i 5 fs [201-203]. The discrepancy can be reconciled by introduction of an additional lifetime T~ = 2/(7- -1 - Ti-l ) attributed to elastic scattering processes. Here, T 1 is the energy relaxation time and T~ accounts for dephasing of the image state e.g. by elastic scattering [199]. The elastic scattering contributes to the observed line width of the energy spectrum but not to the lifetime measurement which measures the population of the image state [204]. Relaxation times of electrons in the energy range 0-2 eV above the Fermi level have been studied recently on Cu and Ag in comparison to Ta [206]. The lifetime of these hot electrons in Ta is 4-7 times shorter than in the noble metals. This is explained by the high density of states at the Fermi energy of Ta. Deviations in the energy dependence of hot electron lifetimes on copper from the parabolic prediction of the Fermi-liquid theory have been discussed in [207]. Ref. [208] demonstrates that the dynamics of a photoexcited electron depends critically on the history of its generation process, i.e. whether the initial state of the electron originates from a d- or sp-band. Further measurements on excited adsorbate states are found in [209]. Recently, an interferometric two-photon photoemission technique has been developed, which allows measurements of polarization dynamics at a metal surface

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with energy, momentum and phase resolution [210]. It has been applied to the Shockley surface state on Cu(1 1 1). Experiments using energy- and time-resolved 2PPE simultaneously can help to distinguish between different contributions to photoemission line widths. An interpretation of results from image states can probably be extended to the occupied Shockley-type surface states. Further interesting results are expected for the near future in this field.

5.

Summary and conclusions

In the last years experimental energy- and angular-resolution in angle-resolved photoemission have been improved considerably. The most advanced electron spectrometer presently available can reach a resolution of AE < 3 meV and A0 = +0.2 ° [14]. Additionally, the line width of the exciting radiation emitted by modern high-intensity light sources was improved in the last years for discharge sources (Ahw < 2 meV) [10] as well as for monochromatized synchrotron radiation. With new generations of high-brilliance synchrotron radiation sources a line width of Ahw < 1 meV is intended. With these excellent experimental conditions it will become very interesting to study photoemission line shapes systematically. The measured line width is no longer dominated by instrumental resolution but it may reflect important solid state properties of the sample. In some case studies on noble-metal surfaces we have discussed the effects on the line shape of bulk and surface state emission peaks due to photohole and photoelectron lifetime, electron-phonon interaction, defect and impurity scattering and, of course, the experimental resolution parameters. For bulk-emission peaks at sample temperatures of T << OD the line shape may be explained almost exclusively by lifetime broadening of both the photohole and the photoelectron final state. Even nonLorentzian line shapes have been modeled in good agreement to the experiments. Furthermore, it was demonstrated that the bulk-emission spectrum is not influenced by scattering of the photoelectrons at step edges. In contrast, at elevated temperature (T > OD) the photoelectrons are scattered strongly by phonons. Spectra taken from noble metals above room temperature are dominated by phonon-assisted transitions. A quantitative interpretation of line shapes at high temperature has not been realized yet, since multi-phonon scattering processes must be taken into account, and it is not known how to implement this properly within the existing photoemission theories. With the improved understanding of photoemission line shapes at low sample temperature an analysis of experimental direct transition intensities has become possible even in the case of strongly overlapping d-band emission peaks. We have compared experimental line intensities to one-step calculations, because this is a very sensitive test for the quality of the wave functions used for the calculation of matrix elements. The overall agreement is found to be unsatisfying. More systematic studies of matrix elements and of the surface optical properties are required for a better understanding of the observed discrepancies. An interesting question concerning the hole lifetime has been discussed for a long time: Does the inverse hole lifetime Fh vanish in the vicinity of the Fermi energy? The Landau theory of Fermi liquids predicts that /'hoc (EF -- El) 2 if lifetime is limited by Auger-decay. However, experimental evidence for this energy dependence in the vicinity of EF could not be resolved on the noble metals up to now. For photohole lifetime studies in particular surface state emission peaks were used, since their line width is not affected by the photoelectron lifetime. The line width of surface state emission peaks is considerably smaller than that of bulk-emission peaks. Recently, line widths of FWHM < 30 meV have

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been measured. Along with a high-resolution study of the surface state at F on Cu(1 1 1) we have discussed various contributions to its line width. Due to scattering of the hole state at step edges, defects and impurities [6,142,160,178-185] the photoemission peaks are broadened. This presently makes a measurement of the small inverse hole lifetime according to Auger-decay impossible in the vicinity of EF. We have also discussed experimental evidence that measured line widths are severely influenced by the sample quality. An improvement in the preparation of very well-ordered single crystal samples (e.g. with a mosaic spread negligible as compared to the angular resolution) and their surfaces (low densities of steps and defects) is urgently required if we want to exploit the very high resolution of modern spectrometers. Not only defect scattering but also scattering of the photohole with phonons limits the hole lifetime. Based on an analysis of the temperature dependence of the photohole lifetime the electron-phonon interaction has been studied. From the slope of !'h(T) an electron-phonon mass-enhancement parameter )~ may be determined. The slope can be measured reliably not only in the vicinity of the Fermi energy, but also at binding energies of several eV. Angle-resolved photoelectron spectroscopy enables us to measure ~ at special k-space points, in contrast to other methods which average over the Fermi surface. By an extrapolation of F h ( T -'~ 0) defect and impurity scattering can be studied. This may become an interesting new field of high-resolution ARUPS, since these scattering processes are not understood quantitatively yet. In conclusion, the analysis of high-resolution ARUPS data beyond the kinematic interpretation, in particular detailed studies of width, shape and intensity of particular emission lines, open a new and very interesting field of surface science, and may enable us to measure various quantifies concerning the properties of the correlated many-electron system.

Acknowledgements I wish to express my gratitude to A. Goldmann for providing continuous support and a scientifically stimulating environment. It is a pleasure to thank G. Meister for many helpful discussions and J. Braun for the possibility to use his one-step photoemission program. Enjoyable collaboration with R. Paniago and E Theilmann is gratefully acknowledged. I thank R. Hennig and G. Lauff for support in constructing electronic and mechanical components. Our work is continuously supported by the Deutsche Forschungsgemeinschaft (DFG).

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