angle-resolved photoemission - theory and current appln. [surface sci.., catal. 74] - s. kevan (elsevier, 1992) ww

PLUMMER In angle-resolved photoemission the study of the energy and angle dependence of the pho- t,oemitted electrons provides information about the electronic excitations of the solid.

Studies in Surface Science and Catalysis 74

Theory and Current Applications

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Studies in Surface Science and Catalysis

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PREFACE

The technique of angle-resolved photoemission (ARP) is at an interesting period in its development In the past 15 years, a theoretical foundation has been laid upon which most current experiments are interpreted: conservation of parallel momentum, approximate conservation of perpendicular momentum, broadening mechanisms, and prediction, detection, and characterization of intrinsic and extrinsic surface states It thus appears that ARP can be applied in a relatively straightforward fashion to a wide variety of problems of current and standing interest in solid state and surface physics and chemistry However, increasingly sophisticated experiments are testing and limiting the application of some of these simple concepts: many body and other final state effects, static and dynamic disorder, theoretical treatment of excitation spectra

In the same period, significant improvements in experimental and theoretical methodology have been attained The techniques for preparing and characterizing surfaces and interfaces have progressed to the point where reasonably complex yet well-defined systems can be prepared: elemental surfaces of all sorts, metal-metal and metal- semiconductor interfaces, semiconductor heterojunctions, compound and alloy surfaces The constant improvement in computer technology and in codes for calculating electronic structure have allowed the "routine" introduction of self-consistency, improved treatments

of exchange and correlation, and relativistic effects The first few steps in actually calculating the excitation spectrum of simple systems have recently been reported Finally, the increased availability and improved quality of synchrotron radiation sources have made the technique more powerful, more generally applicable, and more diverse in the ever- increasing array of sub-fields being spawned The rate at which new storage rings and beam lines dedicated to the production of soft x-rays are being proposed, constructed, and commissioned suggests a very bright and busy future for the technique This confluence of events is allowing ARP to be applied in many laboratories around the world to a variety of systems

This confluence also makes the present an opportune time to produce a research- level monograph on the subject As yet, no comprehensive treatise exists Very good reviews of A R P by Plummer and Eberhardt, Himpsel, and Williams, Srivastava, and McGovern have appeared The several books on photoemission as a whole generally contain but one chapter dealing with ARP None of these reviews, however, comes close to

a comprehensive treatment of this very large and growing field Indeed, it is unlikely that

vi

any one small set of authors would endeavor to write a monograph at the level and in the detail the field warrants What is needed is a reference book that will be of general use both to long-time workers in the field as well as to the uninitiated graduate student just learning how to apply the basics to their particular problem

The first chapter provides an introduction to the motivations, methodologies, and terminologies of the technique, and briefly discusses "the party line" for interpreting ARP data The next two chapters discuss

in detail the physics of the photoemission process and the current understanding of its precise relationship to crystalline electronic structure, primarily €or bulk, three-dimensional states After a brief review of the one-step, single particle theories, these chapters will focus on the "crucial issues" which all-to-often are not adequately addressed in interpreting experimental results These would include, for example, the physics of quasiparticle excitation and other many-body effects, the applicability of the local-density- approximation-calculated electronic structures to photoemission data, and the various contributions to linewidths and shapes The next eight chapters discuss various well-

established and currently active experimental applications of the technique All but chapter 7 are focused upon measurement of intrinsic and extrinsic (i.e., adsorption-

induced) electronic states in two and three dimensions Chapters 4 and 5 survey the surface electronic structure of metals and semiconductors, respectively, as probed by ARP, and its impact upon surface stability and reconstructive behavior Chapter 6 discusses

more complex metals and metallic compounds and is included as an avenue to test simple data analysis models Chapters 8-10 center on the application of ARP in studying the electronic and geometric structure of relatively simple atomic and molecular adsorption systems Chapter 11 discusses the somewhat more complex application to thin film systems Chapter 7 is the only one specifically directed toward core-level ARP measurements, wherein ARP can provide valuable surface structural information All of these subjects are quite active in various laboratories around the field The final chapters examine applications which are still being developed and which hold significant promise for the future Chapter 12 reviews the application to ferromagnetic systems, an area which has been revolutionized by the ability to distinguish the spin of the excited electron at arbitrary energy and emission angle Chapter 13 is included to demonstrate the time-reversed

application of ARP, inverse photoemission, which, as a complement to ARP, allows the

unoccupied levels to be probed The next chapter reviews recent efforts to apply pump- probe techniques, using lasers as the pump, to study the dynamical properties of surfaces in real time Finally, chapter 15 discusses the most recent and perhaps most dramatic application of ARP to highly correlated electronic behavior

This is the goal of the current monograph

N.V Smith and S.D Kevan

The Physics of Photoemission 15

J.E Inglesfield and E.W Plummer

Quasiparticle Excitations and Photoemission 63

S.G Louie

Surface States on Metals 99 S.D Kevan and W Eberhardt

Surface States on Semiconductors 145

G.V Hansson and R.I.G Uhrberg

Metallic Compounds and Ordered Alloys: Carbides and Nitrides,

Applicability of Simple and Sophisticated Theories to More Complex Systems 213

L.I Johansson, and C.G Larsson

H.-J Freund and M Neumann

Metallic Films on Metallic Substrates 371

K Jacobi

viii

Thin Films on Semiconductors 435

R.D Bringans

Spin- and Angle-Resolved Photoemission from Ferromagnets 469

E Kisker and C Carbone

Inverse Photoemission 509 P.D Johnson

Multi-Photon Photoemission 553

J Bokor and R Haight

New Frontiers: Highly-Correlated Electronic Behavior 571 R.F Willis and S.D Kevan

Future Prospects in Angle-Resolved Photoemission 595 S.D Kevan

J.E Inglesfield, University of Nijmegen, Faculty of Science, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands, and E.W Plummer, Department of Physics, David Rittenhouse Laboratory, University of Pennsylvania, Philadelphia, PA 19104-6396

3) Quasiparticle Excitations and Photoemission

S.G Louie, Department of Physics, University of California, Berkeley, CA 94720

S.D Kevan, Physics Department, University of Oregon, Eugene, OR USA 97403, and W Eberhardt, Institut fur Festkorperforschung, Kernforschungsanlage Julich GmbH, Postfach

1913, D-5170 Julich, FRG

5 ) Surface States on Semiconductors

G Hansson and R Uhrberg, Department of Physics and Measurement Technology,

Linkoping Institute of Technology, S-581 83 Linkoping, Sweden

6 )

L.I Johannson, Department of Physics and Measurement Technology, Linkoping University,S-58 1 83 Linkoping, Sweden; and C.G Larsson, Department of Physics, Chalmers University f Technology, S-41296 Goteborg, Sweden

Metallic Compounds and Ordered Alloys

H.J Freund, Lehrstuhl fur Physikalische Chemie I, Ruhr-Universitat Bochum, Postfach 10

2148, 4630 Bochum 1, FRG; and M Neumann, Fachbereich Physik, Universitat Osnabruck, Barbarastrasse 7, 4500 Osnabruck, FRG

11) Thin Films on Semiconductors

R.D Bringans, Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, Ca

94304

12)

E Kisker, Institut fur Angewandte Physik, Universitat Dusseldorf, Universitatstrasse 1,4000

Kernforschungsanlage Julich GmbH, Postfach 1913, D-5170 Julich, FRG

Spin- and Angle-Resolved Photoemission from Ferromagnets

13) Inverse Photoemission

P.D Johnson, Physics Department, Brookhaven National Laboratory, Upton, N.Y USA

11973

14) Multi-Photon Photoemission

J Bokor, AT&T Bell Laboratories, Crawfords Corner Road, Holmdel, N.J USA 07733, and

R Haight, IBM T.J Watson Research Center, P.O Box 218, Yorktown Heights, N.Y USA

10598

15)

R.F Willis, Physics Department, 104 Davey Building, Pennsylvania State University,

University Park, PA 16802, and S.D Kevan, Physics Department, University of Oregon,

Eugene, OR USA 97403

New Frontiers: Highly Correlated Electronic Behavior

16)

S.D Kevan, Physics Department, University of Oregon, Eugene, OR USA 97403

Future Prospects in Angle-Resolved Photoemission

Chapter 1

INTRODUCTION

From humble beginnings in the early 1970's, angle-resolved photoemission spectroscopy (ARPES) has become established as an indispensable tool for the investigation of solids and their surfaces This book represents an attempt to assemble in one volume an account of the large variety of work now going on This opening chapter sets the work against the larger perspectives of the history of the photoelectric effect and of the electronic structure of condensed matter It offers also a brief treatment of past and present experimental methods, and a brief account of our current understanding

1 HISTORICAL BACKGROUND

1.1 Prehistory

Interest in the angular dependence of the photoelectric effect can be traced back to the early decades of this century Jenkin (1) has written an entertaining and informative history which covers this period, and he documents how a number of Nobel laureates (W

H Bragg, C T R Wilson, A H Compton, W Bothe, C D Anderson and E 0 Lawrence) contributed to this topic before moving on to other (and evidently more rewarding!) endeavors The history by Jenkin confines itself to the angular dependence of X-ray photoemission We attempt here to fill in some of the gaps relating to ultraviolet photoemission and its angular dependence Our treatment is not exhaustive, but is intended rather to sound a few historical keynotes which resonate strongly with current activity

In the 1920's, the angular dependence of photoemission from alkali metals was investigated by Ives and coworkers at the Bell Telephone Laboratories (2) Their

apparatus is shown in Fig 1 These pictures exemplify not just the delightful scientific artwork of an earlier generation but also the two main experimental approaches still in use today: a single movable electron collector, or a sectored collector The work of Ives and his

group was closely linked with their technological interest in the use of alkali-based photocathodes in the emergent industry of television and in the possibilities of videotelephony One question of physics raised in this work, however, has lost none of its savor in the intervening decades, namely, the vectorial photoeffect, which is concerned with the differences in emission intensity associated with the polarization of the incident radiation

The next landmark occurs in 1945 with the publication by Fan of a theory of the bulk origin of the photoelectric effect (3) This paper, which does not appear to have had

2

F

I

b

Fig 1 Early angle-resolved photoemission apparatus of Ives and coworkers reproduced

from Ref 2 The method on the right employs a moveable electron collector; that on the left employs a stationary sectored collector

much impact at the time, presented a view of the photoemission process contrary to the

prevailing notion that the photoelectric effect was a surface phenomenon (4) The Sommerfeld model of a metal treats electrons confined in a potential well V(r) of

rectangular shape Optical excitation occurs only if VV#O, and this condition, in the Sommerfeld model, occurs only at the surface If we allow the existence of some atomic structure within the well, we have VV#O in the interior and the existence of a bulk contribution to the photoelectric effect The Fan paper treats the bulk potential by Fourier synthesis, what in modern parlance we would call a nearly-free-electron (NFE) or

pseudopotential model Figure 2, reproduced from the Fan paper, shows the k-space

geometry for optical excitations within a hypothetical NFE metal We recognize here a number of results which have subsequently been rederived by others (5,6) Surfaces of constant photon energy are planes Surfaces of constant electron energy are spheres which intersect these planes Thus the angular distribution of photoelectrons will be about cones

(6)

1.2 Photoemission as a Soectroscooy

The transformation of photoemission into a spectroscopy, as opposed to an interesting and useful physical phenomenon, took place some time in the late 1950s, or early 1960's The contributions of Spicer (at Radio Corporation of America, and later at Stanford University), Apker, Taft and Phillip (General Electric) and Gobeli and Allen

ll

V

Fig 2 Diagram reproduced from Fig 1 of the 1945 paper by Fan (Ref 3) showing the k-

space geometry for the bulk photoelectric effect

(Bell Labs) are especially important Parallel efforts were being made in Europe by Mayer and associates (Clausthal) Some future historian of science might wish to note that photoemission research in the United States appears to have been driven not so much by the desire for fundamental knowledge for its own sake but by the imperatives of the burgeoning television industry

The key discovery in this early period was the establishment of the primacy of the

bulk photoelectric effect The personal memoir of W E Spicer on his early days at RCA

(7) is particularly revealing on this point He was confronted at the start of his work by a

large literature of photoemission experiments performed in ill-defined vacuum using an interpretive approach dominated by the Sommerfeld model This body of work he found

"basically useless" The historical turning point came with the routine attainability of ultrahigh vacuum and the ability to prepare samples which were atomically clean and the availability of bulk band structure calculations for purposes of comparison

A major landmark was the publication in 1964 by Berglund and Spicer (8) of

photoemission energy spectra on Cu and Ag These spectra displayed in a spectacular way the edges of the d bands at respectively 2 eV and 4 eV below the Fermi level The sight of

4

these spectra convinced one of the authors (NVS), then a graduate student, that he wanted

to be a photoemission spectroscopist He was not alone in this aspiration There followed

an explosive effect to use photoemission in the determination of the densities of states and other electronic properties of a wide variety of materials The reader is referred to the compendium by Cardona and Ley (9) for a summary of this activity up to about 1977

In the early 1970's, photoemission began to diversify There was a reawakening in the interest in the angular dependence of photoemission (see Section 1.3 immediately following) The attractive features of synchrotron radiation were also recognized (10, 11)

Spin asymmetry in photoemission was detected (12) Surface effects in photoemission,

having been in eclipse for a decade, now began to reassert themselves Band-gap surface states were observed on clean silicon (13, 14) Electronic states associated with adsorbed molecules were observed (15) Even the elusive surface photoelectric effect was unambiguously isolated (16)

1.3 Angle-resolved Dhotoemission sDectroscoDy

Photoemission work in the 1960's was almost exclusively angle-integrated An

exception was the work of Gobeli, Allen and Kane in 1964 (17) In a notably prescient paper, Kane argued that the E(k) band structure could in principle be mapped from angular dependent photoemission spectra (IS) This paper recognizes the indeterminacy of

kL, the internal perpendicular component of the electron wave vector, and contains within

it the energy-coincidence strategy for overcoming this obstacle Ten years were to elapse however before a band structure was actually mapped (19)

Experimental work on the angular possibilities of photoemission spectroscopy started in earnest in the early 1970's Using a sectored-collector apparatus similar to that

in Fig 1, Gustafsson et al (20) showed in 1971 that the photoemission from Ag(ll1) was indeed distributed about cones of constant energy, as anticipated in the work of Fan (3) and of Mahan (6) At about this time the following events occurred: Feuerbacher and Fitton showed that normal photoemission from W(100) was dominated by a surface state just below the Fermi level (21); Wooten et al demonstrated strong angular dependences in photoemission from GaAs (22); Koyama and Hughey, using synchrotron radiation, observed an angular dependence in photoemission from polycrystalline gold (23); and Williams et al found that the photoemission spectra from MoS2 varied in a spectacular fashion with angle of emission (24) The work of Wooten lends itself to an interesting anecdote At that time, he was at the Livermore Laboratory, and the underlying motivation for his work was the need to develop better photodetectors to monitor emissions from underground detonation of nuclear devices (25)

The first formal demonstration of band mapping using ARPES was published by Smith, Traum and DiSalvo in 1974 (19) In order to circumvent the indeterminacy of k L

these workers performed their measurements on the two-dimensional layer-compounds TaS2 and TaSe2 They monitored the variation in energy E of peaks in the photoemission

5

spectrum with polar angle 0 of emission, and then obtained the parallel component of the electron wave vector using

k 11 = ( 2 1 n E / f i ~ ) ~ / ~ sin 8,

The resulting E(k11) dispersion curves were in good agreement with the first principles

band calculations (26) Equation [l] is now the standard algorithm in the reduction of

angle-resolved photoemission data The use of synchrotron radiation to enhance the capabilities of band mapping and to identify wave function symmetry using polarization selection rules was soon established (see below) The work of this era is captured in the compendium by Feuerbacher, Fitton and Willis (27) A number of more mature reviews are also available (28-31)

Following this hesitant start, ARPES has burgeoned into a major industry Activity shows no sign of slackening Subsequent chapters of this book represent an attempt to organize and to summarize this large body of material

With some qualifications, there is now a general consensus on the physics of the photoemission process and on how ARPES data should be interpreted This has been the

subject of extensive experimental and theoretical work in the past 20 years Indeed, these

issues were the primary focus of previous monographs and reviews of photoemission which can be found in the literature The modern extensions pertaining to the theoretical foundations of ARPES can be found in the next two chapters of this book

2.1 Photoexcitation Drocess

(i) Basic Formula ARPES is intimately tied to investigations of the electronic

structure of crystalline systems Except in the case of very high photon fluences (see

Chapter 14), the process is very well described by lowest order time-dependent

perturbation theory and thus by Fermi's Golden Rule, derived in Chapter 2:

J=(k/4r2) 1 I (Qf I (e/2mc)(A*P + P.A) I qi) I 6(E-Ei-hw)

i

This expresses the observed photocurrent .at final energy E in terms of the initial and final state many-body wave functions, respectively q i and qf, and the dipole operator of the incident photon field Fermi's Golden Rule provides the essence of the so-called single- step, ultimately quantum mechanical model for photoemission In general, the many-body wave functions are not known In order to understand and to interpret a photoemission experiment at a given energy and momentum, various approximations are made The validity of these, described briefly below, is addressed throughout this book

6

(ii) IndeDendent Darticle awroximation A common approximation in applying [2]

is to assume that the initial and final state electronic wave functions may be approximated

as independent particle states In this case, qi and qf can be written as product functions

of band states By virtue of Bloch's theorem, these can be labelled by their energy and two-

or three-dimensional crystal momentum, depending on the degree of surface localization Since the energy and momentum of the final-state eiectron is measured, the dispersion

relation of the final-state quasiparticle dispersion relations can often be determined A

further approximation is commonly made that these quasiparticle dispersion relations are related to the ground state calculated band structure The validity of these two major approximations is of central concern in the following two chapters

The validity of the independent particle picture must be examined on a case-by-case basis For example, "residual" atomic effects (Cooper minima (32), Fano-like resonances (33), shake-up structures (34) etc.) are commonly observed in photoemission spectra from solids These suggest a higher degree of electron correlation, and thus many-body effects, than the independent particle approximation allows One of the outstanding problems in solid state physics, understanding the coexistence of, and interplay between, localized electron correlation phenomena and delocalized, band-structure effects is currently also a

major focus for ARPES (35) In condensed matter systems the importance of these effects

is significant if the on-site correlation energy between two electrons in a band is comparable to the band width The future of such studies is explored in Chapter 15 One facet of ARPES in which many-body effects can never be entirely neglected is final-state lifetime broadening (36) This damping is of both fundamental and practical interest since it ultimately limits the resolution of the technique ARPES owes its surface sensitivity to the strong inelastic scattering which the final state electron experiences as it leaves the crystal The photoelectron is thus endowed with a finite mean-free-path and lifetime Moreover, photoemission is a final state spectroscopy which measures the energy

of the (N-1) particle system relative to that of the N-particle system The hole states below the Fermi level will also have a finite lifetime due to refilling by radiationless processes Both of these lifetimes are of order seconds, so that the loss of energy resolution due

to uncertainty broadening can be substantial In the spirit of Fermi liquid theory, these effects are often treated heuristically by allowing the self-energies of the final state

quasiparticles to be complex (see Chapters 2 and 3) The imaginary parts are then

inversely related to the quasiparticle lifetimes The use of complex self-energies appended

to electron-energy-band calculations is not rigorous, nor is it theoretically satisfying The above discussion indicates that photoemission spectra cannot be accurately compared to ground state calculations in any case Recent theoretical advances are allowing quasiparticle spectra to be calculated directly (37) These advances and their impact upon

the analysis of ARPES data are examined further in Chapter 3

(iii) Surface Photoeffect The surface photoelectric effect arises when the dipole

operator in the Golden Rule is transformed into a gradient of the electrostatic potential

using the commutation relation between the momentum operator and the unperturbed Hamiltonian The difference between the bulk and surface photoeffects has become blurred since it is now clear that both can exist in the same spectrum It is generally accepted that the "original" surface photoeffect which is produced by the rapid potential variation near the surface, is most easily measurable in simple metals with very weak bulk pseudopotential While this was first suggested from total photoyield experiments (16), it

has been usefully studied more recently in simple metals using the polarization dependence

of the photoemission cross section at photon energies near the plasma frequency (38)

2.2 Phenomenology

The manifold of angular parameters in a modern photoemission experiment is

illustrated in Fig 3 Most important are 8, and 4,, the polar and azimuthal angles of electron emission relative to the sample normal and the crystal axes

Fig 3 All the angles This diagram is intended to shown all the angular parameters of a

fully characterized photoemission experiment

Other angles are oP and $,, the polar and azimuthal angles of photon incidence The degree of polarization of the incident radiation is also significant and is generally expressed

as a ratio between amplitudes of electric vector perpendicular (s-polarization) and parallel (p-polarization) to the plane of incidence Circular or elliptical polarization

corresponds to a phase angle A between the s and p components Finally, we recognize

the possibility of a spin asymmetry of the emitted photoelectrons, up or down relative to some appropriately chosen spin-quantization direction

No experiment, as far as we are aware, has had variational control over all of these angular and directional parameters The typical experiment confines itself to some subset

of these angles depending on the particular physical phenomenon under investigation Indeed, the selection of subsets serves as a convenient way to categorize the area of study

band mapping, photoelectron diffraction, symmetry, spin detection and so on

(i) SamDle Orientation The sample in an ARPES investigation is generally a single crystal of known orientation and of high surface quality In the case of semiconductors or layered compounds, the surface can be produced by cleavage in vacuum

In the case of most metals and those semiconductor surfaces not achievable by cleavage, a nearly perfect surface may be produced by appropriate cycles of ion bombardments and annealing, or in some cases by vapor deposition film growth The conditions of surface cleanliness and surface order are established using in situ Auger spectroscopy and low energy electron diffraction (LEED) It is now routine to create ordered overlayers of adsorbed atoms and molecules on these clean surfaces

(ii) Band MaDDing The principal angles of concern are 8, and de, the take-off angles of the photoelectrons The polar angle 8, determines the parallel momentum k in the crystal azimuth defined by de Herein lies the basis of the bandmapping capability of ARPES This is a vast topic which w l be pursued extensively in the following Chapters

II

(iii) Photoelectron Diffraction The emphasis in photoelectron diffraction (PhD) is

on the determination of atomic structure rather than electronic structure The basic notion

is to excite electrons out of core levels and to examine the angular distribution The diffraction patterns observed should, in principle, reveal the environment of the emitting atom The feasibility of PhD was demonstrated in 1978 (39-41) The topic has now reached considerable maturity, and is treated in Chapter 7 A review by Fadley is also available (42) There are two basic choices of angular variable One is to hold 8, constant (usually at normal emission, 8, = 0) and to monitor the core photoemission cross section a function of energy E by exploiting the continuum nature of synchrotron radiation The other approach is to hold ee constant at some off normal Be P 0 position and to measure the azimuthal (4,) dependence of the cross section by rotating the sample

(iv) Svmmetry considerations The direction of incidence of the photons is

specified in Fig 3 by the angles eP and $y Of more significance is the state of polarization

of the incident beam If the incident beam is linearly polarized, we may distinguish between s and p polarization depending on whether the electric vector is perpendicular or parallel to the plane of incidence

9

The photon polarization enters into the cross section through the square of the

momentum matrix elements as indicated in [ 2 ] The final state $f is a plane wave at the

detector, so we may infer something about the angular dependence of the wave function of the initial state $i by variations of A, the electromagnetic vector It should be emphasized that a quantitative treatment is quite difficult since A changes from its exterior value to its value inside the solid over a length scale comparable with the sampling depth of the

photoemission experiment (43) Many applications of [ 2 ] , however, are qualitative, and are

concerned with identifying odd or even symmetry for the initial state wave function (44)

(v) SDin asymmetry There is a class of experiments which measure the spin-

polarization of photoemitted electrons In such experiments, we must specify a direction of spin quantization There are two basic physical origins for spin asymmetry The first is relativistic effects (i.e spin-orbit interaction) whose detection requires circularly polarized light; the appropriate direction of spin quantization is either the surface normal or the propagation direction of the incident photons The second is exchange (i.e magnetic) effects; the appropriate direction of spin quantization direction is the applied magnetic field These matters are elaborated in Chapter 12 The reader is referred also to the

chapters on photoemission in the books by Kirschner (45) and by Feder (46)

(vi) Inverse Dhotoemission The early 1980's witnessed the emergence of

angle-resolved inverse photoemission The inherent cross section for inverse photoemission is lower than that for forward photoemission by the ratio r = (A \A )2, where xe and are respectively the wavelengths of the photoelectron and photon In the

P

ultraviolet region, we have r- a result which explains the relatively late development

of inverse photoemission The angular variables, however, remain unchanged except, of

course, that the directions of the electron and photon in Fig 4 must be reversed This topic

is treated in Chapter 13 Other reviews (e.g Refs 47 and 48) are available in the

(EMDA), 127' cylindrical deflection analyzer (CDA) and others (see Refs 49 and 50)

The 180 SDA is especially well adapted to angle-resolved photoemission for a number of reasons It can be easily matched to axial input optics composed of cylindrical electron lenses One such design by one of the authors (SDK) (Ref 51) is shown in Fig 4 The

four-element input optics permits the angular acceptance and energy resolution to be

adjusted by externally applied voltages Another attractive feature of the SDA is its

point-to-point focussing and the fact that the output focal surface is plane, which lends itself well to parallel detection using microchannel plates

The SDA is inherently angle-selective, and a crude angle-resolved experiment can

be done simply by tilting a sample in front of a fixed SDA It is now common practice,

however, to mount a modest-sized (typically 50 mm radius) SDA on a one-axis or two-axis goniometer, thereby permitting considerable versatility in the choice of angles of emission Such instruments are commercially available from a number of manufacturers These may

be regarded as the modern-day version of the movable collector approach of the 1920's

illustrated in Fig 1

Fig 4 Layout of a spherical deflection analyzer (SDA), workhorse of the ARPES industry, from Ref 51

de field One can think of this as a modern version of the sectored-collector approach of the 1920's illustrated in Fig 1 Such instruments are really high-pass filters for the electron energy E, and the energy spectrum must be extracted by differentiation of the photocurrent with respect to retarding voltage This necessity is eliminated in the elliptical mirror display analyzer (EMDA) perfected by Eastman and coworkers (53) It consists of sets of retarding grids (high pass filters) and reflecting grids (low pass filters) permitting the selection of a narrow A E band pass

For the purposes of band mapping, a more appropriate pair of variables would be E and 8, The aim of such experiments is to determine the E(se), or equivalently E(k 11 ),

dispersion relations for one or two high symmetry azimuths Thus the azimuthal angle be

Ba8e Plate

U

Fig 5 Layout of the E, ee-multidetecting toroidal analyzer of Riley and Leckey (Ref 54)

Another very noteworthy (E, ee)-multidetecting instrument is the magnetic deflection instrument perfected by Uveque (Ref 55) It permits display of the E(k 11 ) band structure on a fluorescent screen in real time Results obtained on the layer compound

GaSe are shown in Fig 6 This work symbolizes in a rather spectacular way the fulfillment

of the dream expressed 25 years ago by Kane (18) that it should be possible to map the

energy bands of solids directly from experiment

m r m r r m k r k m

Fig 6 ARPES results on the layer compound GaSe by Uveque Ref 55) Upper row of

azimuths d d d l e row: images after processing to enhance band structure effects, and converted to (E,k ) coordinates Lower row: band structure diagrams corresponding to the experimental kimuths

panels: (E, 8 ) images taken in real time on a fluorescent screen 6 or four different sample

3.3 Time-of-Flipht Methods

Time-of-flight (TOF) instruments offer an alternative to deflection instruments in the measurement of electron energy spectra Indeed, a rather early angle-resolving photoemission instrument built by Bachrach, Skibowski and Brown (56) exploited the pulsed time structure of synchrotron radiation to do TOF energy analysis The TOF

instruments come into their own when the main aim is to do time-resolved photoemission

witnessing the development (58) of photoemission instruments capable of TOF energy analysis combined with two-angle multidetection

See Chapter 14 for an elaboration of this topic

J G Jenkin, J Electron Spectroscopy 23, 187 (1981)

H E Ives, A R Olpin and A L Johnsrud, Phys Rev 32,57 (1928)

H Y Fan, Phys Rev 68,43 (1945)

A Hughes and L DuBridge, Photoelectric Phenomena (McGraw-Hill, New York,

1932)

N V Smith and W E Spicer, Phys Rev 188,593 (1969)

G D Mahan, Phys Rev B2,4334 (1970)

W E Spicer, in Chemistry and Physics of Solid Surfaces IV, edited by R Vanselow

and R Howe (Springer-Verlag, Berlin, 1982)

C N Berglund and W E Spicer, Phys Rev 136, 1030 (1964); ibid., 136,1044 (1964)

M Cardona and L Ley, Photoemission in Solids (Springer-Verlag, Berlin, VoI I,

1978, Vol II 1979)

D E Eastman and W D Grobman, Phys Rev Lett 28,1327 (1972)

G J Lapeyre, A D Baer, J Hermanson, J Anderson, J A Knapp and P L Gobby, Solid State Commun 15, 1601 (1974)

U Banninger, G Busch, M Campagna, and H C Siegmann, Phys Rev Lett 25, 585 (1970)

L F Wagner and W E Spicer, Phys Rev Lett 28, 1381 (1972)

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Chapter 2

J.E INGLESFIELD AND E.W PLUMMER

In angle-resolved photoemission the study of the energy and angle dependence of the pho- t,oemitted electrons provides information about the electronic excitations of the solid In the simplest picture, these excitat,ions correspond to the one-electron states of band theory, and much of this volume is concerned with the invaluable information about surface and bulk band structure which can be found from the photoemission spectrum Many features in the spec- trum correspond to surface and bulk processes Surface photoemission comes from electrons

in the top few angstroms of the solid, and if the surface is periodic it conserves the component

of the electronic wave-vector K parallel to the surface As the energy of the outgoing electron

equals the energy of the initial state plus Rw, the dispersion of surface states, for example,

can immediately be mapped out from the photoemission spectrum In bulk photoemission on the other hand, the component of the Bloch wave-vector perpendicular to the surface kl is

also conserved (approximately) in the transition and the bulk band structure can be mapped out Actually, the surface plays an integral role in photoemission - t h e electrons have t o pass through the surfa.ce on their way to the detector, and the mean free path of these electrons is

rather short inside the solid, typically around 10 for photoelectrons with a kinetic energy

of 50 - 100 eV These effects must be included in an accurate description of photoemission

I n this chapter we shall concentrate a n the way that one-electron energy bands show u p in photoemission, in spite of the complicated electron-electron interactions in solids These bands really describe the excitations of quasiparticles - screened electrons or holes - and much of the current interest in photoemission (and iiiverse photoemission) is centred on the differences between the quasiparticle states, and the energy bands of conventional density functional theory The electron-electron interaction can also lead to extra features in the photoemission spectrum - satellites - and we shall see how these can originate Finally we turn to the electromagnetic field itself, discussing its screening by the electrons

2.1

In angle-resolved photoemission, the energy distribution of electrons travelling in a particular direction is measured To see how this measurement is related to the electronic structure of

the solid we must first understand the wave-function of this final state [l, 2, 3, 41

Switching on the light in the remote past, the wave-function of an electron initially in state

q5* is given a t t = 0 by the perturbation theory expression [5]:

G is the Green function for the system, and the perturbation is:

2nzc with frequency w We shall study the form of t,he vector potential A in greater detail in section

6 As the electrons reaching the detector are free, let us express G in terms of the free electron Green function Go using Dyson’s equation [ 5 ] :

where T is the operator describing the scattering of the emitter The free electron Green function is given in Hartree atomic units ’ by:

where k2/2 is the energy of the emitted electrons, E = E, +FLU As we measure the photocur-

rent a long way from the emitter this can be expanded:

exp(ikr) GO(r,r’) - -~ exp( -ik.r‘),

This wave-function contains the physics of the famous three-step model [S]: working from the

right, the photoelectron is excited by SH, scattered by the crystal, and then propagates to the detector We can rewrite this as:

where:

I4f) = (1 + G 3 ’ ) I4f)

So the photocurrent per unit solid angle is finally given by:

This expression for the photocurrent corresponds to the Golden Rule [2, 3, 71, with a final state wave-function given by (10) This is the time-reversed LEED state [3]: the final state is obtained by shooting the plane wave e x p ( 4 k r ) a t the sample, letting the sample scatter it via (1 + GOT), and finally taking the complex conjugate This is not particularly mysterious, because in photoemission the electron which reaches the detector was scattered by the atoms

in the emitting sample in the distant past, whereas in most scattering problems this takes place in the future The energy density of these final states is given by k/8a3 per unit solid

= h = 1 a u

1 7 angle, and when multiplied by the Golden Rule factor of 27r we immediately recover (11) The Golden Rule expression can be generalized t o the case where there are many occupied electronic states in the emitter, giving for the photocurrent per unit solid angle and per unit energy:

As we shall see throughout this book, a great deal can be learnt from photoemission spectra

by applying conservation rules [S, 9, lo] The first of these - energy conservation - is ensured

by the &function in the Golden Rule expression (12):

E = E; + hw

- from the measured energy E of the photoelectrons and the photon energy hw we can imme- diately deduce the energy of the initial state E, Moreover, in photoemission from a periodic solid surface, t h e wave-vector component parallel to the surface K of the initial and final states is equal t o within a surface reciprocal lattice vector; this is because the vector potential entering the matrix element in (12) varies comparatively slowly (at least parallel t o the surface

- see section 6) As surface states have a discrete energy a t fixed K , they show up as sha.rp features in the angle-resolved photoemission spectrum [figure I ) , and by applying these two conservation rules their dispersion ca.n be mapped out [S 101 (section 4 )

Inside the solid, the LEED state ($;) corresponding to shooting exp(-ik.,r) at the surface consists of the linear superposition of bulk solutions of the Schrodinger equation, with energy

E and wave-vector component K , which matches onto the incident wave over the surface

In general these solutions are Bloch waves travelling away from the surface corresponding to

the energy bands, together with evanescent waves decaying into the crystal from the surface [11]; in an energy gap of the bulk band structure the wave-function is made up entirely of evanescent waves These evanescent waves are solutions of the Schrodinger equation which are not allowed in an infinite crystal, but which can occur in the case of a crystal with a surface

We would then expect the matrix element in (12) t o be large for bulk initial states with the same total wave-vector ( K , k,) as a travelling wave component of the final state, giving a direct transition By measuring the energy of direct transitions as a function of photon energy, the initial state bands can then be mapped out - if the perpendicular component of wave-vector inside the solid, k 1 , can be determined [S, 10) (section 4) T h e presence of the surface means,

of course, tha.t k l is not strictly a good quantum number except deep in the solid, and this

is why the final state (and initial state) wave-functions contain evanescent wave components near the surface So the photoemission spectrum also reflects the local density of states at the surface with fixed wave-vector component K - the continua of bulk states reflected by the surface as well as the discrete surface states [ l l ] [figure 1) In fact, even the travelling wave component of the final state wave-fiinction is damped by many-body effects, giving a finite

mean free path (section 3.2), and this has the effect of smearing out direct transitions (section

3.4) and enhancing surface sensitivity

Transitions to the final state

(13)

I I I I I I “ “ I

-20 -18 -16 -14 -12 -10 - 8 -6 - 4 - 2 0

INITIAL STATE ENERGY ( e V )

Figure 1: Normal emission photoemission from Mg(0001), AI(001) and Be(0001) [9] Direct

bulk transitions are indicated by the arrows, and surface states by the shading

2.3 Calculating photo emission

A calculation of the photoemission spectrum can help with the identification of features as

either surface or bulk, and by “tuning” t,he potential felt by the electrons t o obtain optimal agreement with experiment, information can be obtained about the energy shifts and broad-

ening effects due to the electron-electron interactions (section 3) The necessary ingredients

of such a calculation are an accurate way of finding the electronic states both in the bulk and at the surface, a proper evaluation of the matrix elements, and a way of putting in the many-body effects of lifetime broadening for the initial state and the finite mean free path of

the photoelectron (sections 3.2, 3.3) A computer package t o do this was developed by Pendry and his co-workers [la]

The starting point is to rewrite the Golden Rule expression for the photocurrent using the following relationship between the sum over states i in (12) and the Green function [5]:

This expression is very convenient, because it involves the Green function for t h e initial states,

rather than the individual states themselves In evaluating (15) on the computer, several

approximations have t o be made First it is assumed that the vector potential in SH is

spatially constant, neglecting the screening effects w e shall discuss in section 6 This means that 6 H can be transformed to VV form [12], where V is the potential felt by the electrons

It is assumed that this potential has the muffin tin form inside the solid, that is, a spherically symmetric atomic-like potential at each atomic site, and a flat potential in the interstitial region between the atomic muffin tins This form of potential gives the electronic states very well for reasonably close-packed systems, but is not satisfactory for open structures like diamond At the surface, a simplified (one-dimensional) form of the surface barrier is used

- usually a step barrier, though recent work uses a barrier taken from self-consistent surface

electronic structure calculations [13]

With these simplifications, the way that the program works is as follows The photoelectron state $r, the time-reversed ( i e complex conjugated) LEED state, is calculated using a layer scattering approach in which the solid is divided up into layers of atoms is expanded

in plane waves between the layers, and the reflection and transmission properties of each layer are calculated, giving the probability amplitudes for a plane wave with parallel wave- vector component K t o be reflected and transmitted into plane waves K + G, where G

is a two-dimensional layer reciprocal lattice vector By repeated reflection and transmission operations, the full wave-function for an electron incident on the whole semi-infinite crystal can

be determined Knowing $,, J dr’G(r, r’; E - tLw)6H$,(r‘) can be found In this expression,

SH$f acts as a source of electrons for which G describes the propagation in the lower energy (initial) state An immediate simplification is that 6 H (2.e V V ) is non-zero only inside the muffin tins and at the surface barrier, and then the layer-adapted multiple scattering technique can be used to find the whole wave-field of J dr’GbH$,

As an example of photoemission calculations, figure 2 shows results calculated by Konig et

a1 [13] for normal photoemission from Ag(001), at a range of photon energies, compared with

experiment [14] These were obtained using theii extension of the Pendry program to include the more accurate surface barrier There is fairly good agreement with experiment, and the main features in the spectra are reproduced by the calculation: in particular, this work clearly

identifies state B in figure 2(a) as a surface state The main discrepancy is that the peaks lie about 1 eV closer to t h e Fermi energy than in experiment, due to shifts in the quasiparticle energy bands compared with the density functional values (section 3.3.1)

Many aspects of photoemission can be described in a single particle picture, as a transition from

an occupied one-electron orbital t o the state describing t h e propagation of the photoelectron

In reality the solid is an immensely complicated many-body system of electrons all interacting with one another, but for many purposes this simply results in the screening and decay of the hole left behind and of the photoelectron These screened, decaying single particle states are the quasiparticles [15, 161, whose wave-functions satisfy a single particle Schrodinger equation:

containing t h e self-energy (or optical potential) C, which is complex and energy-dependent The real part of C describes the screening of the quasiparticles, shifting t h e energy from the Hartree value, and the imaginary part the decay or inelastic scattering processes The quasiparticle wave-functions (or amplitudes) are given by the matrix elements:

#(r) = ( N - 1, i\&r)lN,O), hole states 4(r) = (N,Ol&r)lN + l , j ) , electron states (17) Here is the electron annihilation operator, and the hole quasiparticle describes the hole in the ith excited state of the ( N - 1)-electron system, and the electron quasiparticle describes the extra electron in the j t h state of the ( N + 1)-electron system The initial state energy bands measured in photoemission are really the bands of the hole quasiparticles The imaginary part

of the self-energy, which corresponds t o the decay of the hole, gives each state a finite energy width (161 The imaginary part of C felt by the photoelectron comes from inelastic scattering

processes which cause the spatial decay of the quasiparticle amplitude into the solid, away from the surface [16] The decay length (+2 to give the decay in intensity) corresponds to the mean free path, one of the factors which determines surface sensitivity in photoemission

To understand the quasiparticle properties of the final state more fully, let us go back

to the many-body Golden Rule (section 2.1) [17] Writing the matrix element between the many-body states as:

j However, the matrix element describing the photoelectron (k; N - l,s]$+(r)lN - 1,j)

is not an ordinary quasiparticle amplitude as in (17) An electron quasiparticle amplitude would be (k; N, Ol~+(r)lN,O), describing the propagation of an electron asymptotically in the (time-reversed) plane-wave state k incident on the N-electron system in the ground state; in

(k; N - l , s J $ + ( r ) l N - l , j ) , the electron is incident on the ( N - 1)-electron system in an excited state s, and changes the state from to j

Fortunately, the matrix element (k; N - l , s J G + ( r ) J N - l , s ) , in which the photoelectron does not change the state of the ( N - 1)-electron system, behaves like t h e usual quasiparticle amplitude The term j = s in (19) is the intrinsic contribution to the photoemission matrix

element [l], for which the photocurrent can be written as:

shall see later in this chapter and elsewhere in this volume, A describes satellites as well as quasiparticle states associated with single particle energy bands The intrinsic contribution to photoemission dominates at high enough photon energies, so that the photoelectron escapes before the other ( N - 1 ) electrons can respond to it [l]

Let u s now study the photoelectron quasiparticle in detail It can easily be shown that the matrix element appearing in (19) satisfies a system of coupled Schrodinger equations [17]:

x (k; N - 1 , s 1 &+(r”) 1 N - 1,s) =

( E k , s - E , ) ( k ; N - 1,s I &+(r) I N - 1,s)

Here, G is the Green function for the one-electron part of the Hamiltonian, and:

fi,s(r) = /dr’(N - 1 , s /4+(r‘)d(r’) 1 N - l , l ) v ( r ’ , r ) (24) (23) is a n effective single particle equation for the photoelectron quasiparticle amplitude,

containing the self-energy:

This is essentially the same self-energy that appears in (16) [17, 181, evaluated a t the energy

of the emitted electron

23

1 " " " " ' l

Figure 3: Real and imaginary parts of the self-energy of an electron gas at different densities

as a function of wave-vector relative to X.17 [16, 191

When we solve the Schrodinger equation (23) for the (time-reversed) photoelectron, the so-

lution at the energy of measurement E corresponds to a state decaying into the solid [16]

Neglecting band structure effects, the wave-function inside the solid has the form:

+; N exy i k l z exp - 7 2 , (26) and when y is small compared with k l :

So the escape depth is given by:

where k is the total wave-vector inside the solid (more generally, the group velocity)

In a free electron metal, C has a small imaginary part until the electron energy measured

relative t o the Fermi energy exceeds the plasmon energy (figure 3) 116, 191 A new channel

then opens up into which the photoelectron can be scattered by exciting a plasmon In terms

of (as), state 1 consists of the hole state s plus a plasmon, and the imaginary part of the

Energy Above Fermi Level (eV) E n e r g y Above F e r m i Level (eV)

Figure 4: Mean free pat,li a s a function of electron energy in A1 and Ag The lines correspond

to different theories, and the points to experiment [22]

potential comes from (25) when E 2 El - E, Below the plasmon frequency the photoelectron can be scattered out of the elastic cha.nne1 by exciting electron-hole pairs, but we see from

figure 3 that they give a small contrihntion to the imaginary part of the self-energy For

a nearly free electron metal like A1 the mean free path should drop sharply a t the plasma frequency, and then gra.dually increase with increa.sing energy This behaviour is found rather generally, in fact, with a minimum in the mean free path of about 5 A at an electron energy

of typically 20 eV [20, 211 A theoretical analysis of electron mean free paths in a wide range

of elements and compounds has been carried out by Penn [22] and Tanuma e t a2 [23], making

use of optical data for the dielectric function to which the electronic self-energy can be related Results for A1 and Ag are shown in figure 4, together with experimental data from overlayer experiments and photoemission - a.greement, is rather good I n a,ctual photoemission and

LEED calculations, the imaginary part of the self-energy is assumed t o be constant spatially, right up to the surface, and this seems a good a.pproxima,tion [24]

The imaginary pa.rt of the self-energy conies from real transitions scattering out of the elastic cha.nnel, hut C also has a real part coming from virtual transitions These are related

by a Kramers-Kronig dispersion relation [ l G ] For electrons with energies greater than the plasmon energy, the real part of the self-energy gradually becomes less attractive as the electron

energy increases [19] (figure 3 ) This shift comes from the inability of the electron gas t o screen

the photoelectron when it is moving fast There is experimental evidence that this shift is quite important: for example in photoemission calculations for Cu( I l l ) , an upward shift in the inner potential felt by the photoelectron is needed to obtain the right photon energy-dependence of the intensity [ 2 5 ] Beam threshold measurements for the diffraction of low energy electrons from C u ( l l 1 ) show a N 4 eV shift in the real part of the self-energy between 20 and 120 eV kinetic energy 1261, and theoretical a.nalysis of the low energy electron diffraction from Cu(OO1)

over an extended energy range ( u p to 700 eV) [27] exhibits an energy-dependence consistent

with the beam threshold measurements and the theoretical calculations shown in figure 3 The

effect of the energy-dependence of the self-energy can be seen in the inverse photoemission 3xperiments of Speier e t al [as] In these experiments the quasiparticle amplitude for the 2lectron in its high energy initial state incident on the solid is essentially the same as that for the time-reversed photoelectron Speier ef a1 [2S] found upward shifts in spectral features

in the bremsstra.hlung isochromat spectra for Ni, Cu, Ag and Pd relative t o the density of unoccupied states calculated with an energy-independent ground state potential (figure 5)

Figure 5: Bremsstrahlung isochromat spectrum for Ag (top curve), compared with the cal- culated density of states (bottom curve) and the broadened density of states (middle curve)

In (21) we wrote the spectral function in terms of the individual hole excitations of the system, but it is convenient to rewrite this in terms of the interacting Green function or

propagator, upon which the diagrammatic and pcrt.urha.tion approach to many-body theory

is based The Green function is given by:

E < 0 (we take the Fermi energy as our zero), A is related t o G by:

- the same as the non-interacting result (14) The Green function satisfies the inhomogeneous version of the Schrodinger equation (16), containing t h e self-energy [16]:

J

1

2

( - - V * + V ( r ) - E)Q(r,r';E)+ dr"C(r,r";E)O(r",r’,;E) = -6(r-r') (32)

All the difficulties of the many-body prohlein reside in determining C, but gradually experience

is building up in finding the self-energy in simple metals [29, 301 and semiconductors [31], and

photoemission provides an invaluable way of testing the theories

where c is the energy of the state using the Hartree potential, and (C) is the expectation value

of the self-energy If Z varies slowly with energy, and its imaginary part is small, we then find

a quasiparticle peak in A a t energy c + A, lifetime broadened to width r', with A = %C, and

n

r = I&CI [is]:

Such a quasiparticle is a solution of the Schrodinger equation (16) at a particular Bloch wave- vector, with a complex (analytically continued) eigenvalue (c + A + X ) [16] Compare this with the photoelectron quasiparticle which is a scattering solution of (16) at real energy, and consequently complex wave-vector Not all the structure in A corresponds t o quasiparticles, which are usually considered as those states for which there is a one-to-one relationship with energy bands; there are also many-body satellites However, quasiparticle states are particu- larly important,, and a constant theme of recent work is that they are often surprisingly well defined

3.3.1 Core states and s a t e l l i t e s

Photoemission from localized core states shows a shifted and lifetime-broadened quasiparticle peak, and satellite structure The valence electrons screen the core hole left behind, reducing the energy needed t o remove the core electron This interaction between the core hole and the remaining electrons inevitably leads to the possibility of electronic excitations - plasmons in the case of nearly free electron metals, as well as electron-hole pairs - this gives the satellite structure

To study the core hole spectral function, we shall describe the screening properties of the valence electrons in metals like Na or Al in terms of plasmons, writing the Hamiltonian as

a measure of the potential which t,liis produces at the core The eigenstates of (35) fall into two classes Clearly, when the core st;lte is occupied, H just reduces t.o the simple harmonic oscillator for plasmons, with energies:

27

W * E c - 3 W p W + € c - 2 W p W’Ec- W p W ‘ E c kinetic

energy

Figure 6 : Spectral function for a core state i n a free electron gas with the density of Na (rs = 4 )

(displaced by the photon energy hw to give the intrinsic photoemission spectrum) [l]

T h e spectral function d ( w ) calculatd for a core state in a free electron gas with the density

of N a ( r S = 4 a.u.) is shown in figure 6 [I] As well as the no-loss peak a t t, there are the satellites corresponding t o exciting one, two, plasmons, with line-shapes resulting from the plasmon dispersion When the interaction between the core hole and the remaining electrons

is included, the total integrated weight in the spectral function is the same as in the non- interacting case - weight is transferred from the core line to the satellites; moreover, the centre

of gravity of t h e spectral function lies at the unrelaxed core energy c t - the Koopman’s theorem value [33] This is generally true, for more complicated interactions than we have assumed

in (35) Unfortunately these moments results for d do not go over t o the photoemission spectrum except in t h e intrinsic limit of high photon energies, and in the case of plasmon satellites extrinsic plasmon excitation persists even a t high photoelectron energies [l, 17, 341

Although the spectral features remain, their weight changes because of extrinsic effects, in which the excitation of plasmons by the photoelectron becomes important (section 5 )

As well as the plasmons, the core hole can excite electron-hole pairs in t h e electron gas Unlike the plasmons, the electron-hole pairs in a metal start with zero energy, so the corre- sponding satellite begins a t the screened core energy Moreover, the effect of the core hole on the electron gas in its ground state is to modify each one-electron wave-function, resulting in zero overlap between the Sla.ter deterniinant wave-functions of the many-electron system with

and without the core hole [ 3 5 ] This wipes out the 6-function of the core state quasiparticle peak shown in figure 6 It is replared by a power law singularity in the spectral function [l]:

il 1-

Ot-

BINDING ENERGY (OW

Figure 7: Na 1s and 2s photoemission spectra, showing the asymmetric shape of the no-loss peak, fitted by the Doniach-SunjiC lineshape and a Lorentzian a t the surface plasmon satellite (solid line) [3S]

t o quote the famous Nozitres-de Dominicis result [36)

So far we have neglected the lifetime of the hole, due mainly t o Auger processes in which

an electron drops down from a higher level and a second electron is excited If these do not interfere with the interactions with plasmons and electron-hole pairs, the full spectral function can be obtained by convoluting the infinite lifetime result, that is (41) close to ccr

with a Lorentzian like (34) This gives the Doniach-SunjiC lineshape [37]:

where we write f for the gamma-function, and r is the lifetime broadening

In general this provides an excellent fit t o experimental data, and has been applied to a wide range of core states In a study of photoemission from nearly free electron metals, Citrin

et al [38) obtained the core spectra shown in figure 7 for the N a 1s and 2s levels Allowing

for phonon effects, they found t1ia.t their results could be fitted with (43), with a n asymmetry

parameter a of 0.20 and lifetime broadening I? of 0.2s eV for these levels a can also be found from the scattering properties of a static, self-consistently screened core hole via (42), in good

agreement with the Na experiments The lifetimes are harder t o calculate, but for the deep

1s hole i t is mainly due to intra-atomic Auger processes and an atomic calculation gives good agreement The 2s lifetime is dominated by interatomic processes involving valence electrons, but reasonable theoretical estimates can be made for this too

3.3.2 Valence band quasiparticles

Most electronic structure calculations for the valence states in a solid and at a surface are based

on density functional theory, which is designed to give ground state properties correctly, like the total energy and charge density [39] Although the one-electron wave-functions and energies in

density functional theory do not in principle correspond t o quasiparticle excitations, the energy bands are often in good agreement with photoemission measurements The discrepancies will

be studied in section 4.3

In density functional theory, the ground state charge density po(r) of the system is written

in terms of effective one-electron wave-functions which satisfy a Schrodinger equation contain- ing t h e exchange-correlation potential Vzc(r) as well as the Hartree potential V ( r ) and the

potential due to the ions [40]:

V,, corrects VH to describe the effects of exchange - the fact that electrons with the same spin cannot be a t the same place a t the sa.me time - and the correlated motion of the electrons Usually Vzc(r) is calculated in the local density approximation, in which it is taken as the exchange-correlation potential of a uniform electron gas with the same charge density as a t point r Density functional theory is remarkably succesful in describing the charge density and energetics of solids and surfxes, but apart from the highest occupied energy level which equals the ionization energy, the individual t , ’ ~ in (44) have no rigorous meaning as one- electron excitation energies This is unlike the quasiparticle equation (16), for which the c;’s

are the excitation energies which enter the hole spectral function Unlike the self-energy C in

(16), the exchange-correlation potential is local, real, and energy-independent

The discrepancies between density functional energy bands and the actual quasiparticle excitations appearing in (16) are due to energy-dependent energy shifts coming from the real part of the self-energy, and lifetime broadenings due to the imaginary part (34) A particularly important energy shift occurs in semiconductors, for which density functional theory generally gives a gap between t h e valence and conduction bands which is too small [31], but in the following section we will also discuss shifts in met,als The GW approximation [16] provides a

valuable method for calculating C, and this is the way that the free electron results of figure

3 were found T h e reason why density functional theory continues t o be the starting point for describing valence band quasiparticles is that it is relatively straightforward t o implement; the self-energy is much more difficult to determine even in the GW approximation, and up to now it has been calculated only for semiconductors (311 and s-p bonded metals [29]

The lineshape of surfa,ce states, localized states lying in bulk energy gaps, should in prin- ciple be Lorentzian, due to the energy broadening of the imaginary part of the self-energy (section 4.1) [41, 421 Of course the surhce s h t e does not necessarily feel the same self-energy

as neighbouring bulk states, but in the ca.se of 4l(OOl) the surface state below E.P extends

deep into the solid (431, and we would expect the value of the self-energy to be bulklike In

this case, the apparent lineshape of this state is asymmetric (figure 1) [44], because it lies so

close to the bulk band edge that the Lorentzian overlaps with the smeared-out band edge [43]

This does not mean that there is any mixing between the s u r f x e state and bulk quasiparticle amplitudes - the surface state quasiparticle is still spatially localized In the case of Be(0001)

the surface state has the expected Lorentzian lineshape (421 (section 4.1)

The energy broadening of the hole quasiparticle combines with the momentum broadening of

the photoelectron, due to the exponential decay of its wave-function away from the surface,

to give a width to direct transitions To obtain this we go back to the Golden Rule, writing