Metal Optics Near the Plasma Frequency

135 Pages·1986·2.104 MB·English

by  Professor Dr. Frank Forstmann, Professor Dr. Rolf R. Gerhardts (eds.)

E Forstmann .R .R Gerhardts
Metal Optics
Near eht
Plasma Frequency
With 30 Figures
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Professor Dr. Frank Forstmann
Professor Dr. Rolf .R Gerhardts*
Freie Universit&t Berlin, Fachbereich Physik
Institut fer Theorie der kondensierten Materie, Arnimallee 14
D-1000 Berlin 33, Germany
* Present address: Max-Planck-lnstitut f0r FestkSrperforschung
HeisenbergstraBe t, D-7000 Stuttgart 80, Fed. Rep. of Germany
Manuscripts for publication should be addressed to:
Gerhard H6hler
Institut for Theoretische Kernphysik der Universit~t Karlsruhe
Postfach 6380, D-7500 Karlsruhe 1, Fed. Rep. of Germany
Proofs and all correspondence concerning papers in the process of publication
should be addressed to:
Ernst A. Niekisch
Haubourdinstrasse 6, D-5170 J~Jlich 1, Fed. Rep. of Germany
ISBN 3-540-16952-0 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-16952-0 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Forstmann, .F (Frank), 1937-. Metal optics near the plasma
frequency. (Springer tracts in modern physics; 109) Includes index. .1 Metals - Surfaces - Optical properties.
.2 Plasma frequencies, I. Gerhardte, .R (Rolf), .--9491 .1I Title. .11t Series. QC1.$797 vol. 901 530s 530.4'1
27022-68 60.8.671CQ
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Preface
dradnatS textbooks no optics neglect eht fact that light interacting with metal
surfaces or interfaces nac excite longitudinal plasma sevaw inside the metal. This
nonemonehp semoceb more important at frequencies of the order of metal amsalp
frequencies (~ ~ 01 eV), the region in which eht advent of synchrotron radia-
tion sah denepo pu a rapidly growing field of optical experiments. In this emulov
ew demonstrate .pahC( )3 that reflection from dna transmission through metal lay-
ers, electroreflectance dna ellipsometry from metal surfaces, eht surface nomsalp
dna sedomnegie in thin layers dna spheres, dna also the fields near the surface
sa deretnuocne in photoemission cannot eb understood even qualitatively without
taking amsalp sevaw into .tnuocca
A major mia of this koob is to demonstrate that it is ton only necessary to in-
clude amsalp sevaw in eht calculation of the optical response of a variety of sys-
smet containing metal surfaces dna layers, but that it is also possible to od os
within a rather simple, straightforward extension of classical Fresnel optics, the
so-called cimanydordyh" approximation" (HD). This DH sah a clear physical tpecnoc
dna is easy to handle mathematically. In addition to eht familiar transverse elec-
tromagnetic ,sevaw it includes longitudinal waves which era also solutions of
Maxwell's equations if eht rebmun-evaw ecnedneped of the dielectric function, i.e.
"spatial dispersion", is taken into tnuocca in a Drude-type theory. morF a theo-
retical point of view, the DH is eht simplest approximate version of "nonlocal -po
tics". oT discuss its relations to other lacigolonemonehp sa well sa eht erom -neg
eral microscopic sehcaorppa is eht dnoces major mia of this article. It turns tuo
that for ynam sesoprup eht DH provides a very useful interpretation ,emehcs hcihw
allows a qualitative understanding of the underlying physics without requiring a
complicated theoretical dna numerical machinery.
eW therefore want to egaruocne experimentalists to esu the DH for eht inter-
pretation of experiments dna ew start with eht discussion of this slight exten-
sion of standard optics, which sah all the features of a "textbook metal optics".
eW present the dohtem dna demonstrate its successful application in ynam circum-
stances. This emehcs of calculating the optical response of systems with -cudnoc
tion electrons is conceptually on erom difficult than calculation of eht reflec-
tivity of a glass plate dna nac well eb included in graduate lectures no electro-
scimanyd or optics.
ehT dnoces half of this koob sets tuo several erom sophisticated treatments of
the esnopser of a single metal surface to electromagnetic radiation. ehT relations
neewteb general forms of the esnopser functions dna additional yradnuob conditions
era discussed for a variety of lacigolonemonehp models, the ,DH the specular re-
flection model, the "dielectric approximation", eht "semi-classical infinite bar-
rier" model dna several others. Also, microscopic surface response calculations
desab no a mutnauq lacinahcem ledom containing a surface potential for the elec-
trons era presented dna their relationships to eht DH dna eht BICS ledom era il-
lustrated. Finally, ew deal with owt surface response functions d±(m) dna dll(m),
hcihw contain the integrated effect of spatial dispersion dna surface properties no
elbarusaem quantities such sa reflection coefficient or surface nomsalp dispersion.
Their relation to stnemerusaem sa well sa to microscopic or lacigolonemonehp re-
esnops calculations is discussed.
While the dnoces half .spahC( 4 dna )5 of this emulov sesserdda erom the reader
ohw is interested in eht theoretical aspects of eht metal surface response, the
first three chapters era tnaem for ydobyna ohw semoc across optical problems in-
volving conducting surfaces. There should also eb a field of technical applica-
tions: special optical properties of layered structures involving conductors, optimal
or selective mirrors in the synchrotron radiation frequency range dna related prob-
lems. ereH a successful dohtem for calculating the optical properties of gnitcudnoc
smetsys is presented that is simple dna transparent hguone to emoceb a standard
tool in metal optics.
This koob is a result of several years work dna would ton have neeb possible in
this form without interaction with other scientists. eW gratefully rebmemer dna
egdelwonkca discussions with ynam of ruo colleagues in this field, ohw helped su
to clarify eht smelborp dna to reach eht point of view from which ew treat metal
optics in this article. eW would like to mention especially .P Apell, .A Bagchi,
.R Barrera, A.D. ,namdraoB A.M. Brodsky, .A Eguiluz, P.J. ,namlebieF .R ,shcuF
.P Gies, .K ,apmeK K.L. Kliewer, D.M. Kolb, A.S. Kondratjev, A.E. ,amhcuK .S Liu,
.T Lopez-Rios, .S Lundqvist, .T Maniv, D.L. Mills, .G ,yayhdapohkuM .H Raether,
.R Ritchie, .L.W Schaich, .H ,ekhcsnetS .K .mrutS
Finally, ew egdelwonkca financial support from the ehcstueD -niemegsgnuhcsroF
schaft through hcierebsgnuhcsrofrednoS .6
Berlin. Stuttgart, yaM 6891 F. Forstmann . R.R. Gerhardts
lV
stnetnoC
1. Introduction ..................................................... 1
2. Metal Optics ni the Hydrodynamic Approximation .......................... 6
2.1 Plasma Waves, Nonlocality, Spatial Dispersion .................... 6
2.2 The Hydrodynamic Model for Simple Metals ............................ 8
2.3 The Additional Boundary Conditions .................................. 01
2.4 The Additional Boundary Conditions at a Free Surface .............. 21
2.5 The Energy Theorem for Combined Transverse dna Longitudinal Fields .... 41
2.6 The Additional Boundary Conditions at na Interface Between Two Metals
of Different Electron Density ..................................... 71
2.7 Extension to Not-Nearly-Free Electron Metals ..................... 91
3. Applications of Nonlocal Metal Optics ................................... 22
3.1 Reflection dna Transmission ......................................... 22
3.2 Resonances ni Thin Metal Films ....................................... 82
3.3 The Surface Plasmon Dispersion ................................. 92
3.4 Standing Wave Eigenmodes ni Thin Surface Layers ...................... 63
3.5 Optical Properties of Metal Layers no Metal Substrates ................ 14
3.6 Electroreflectance Spectra at Silver Surfaces ........................ 34
3.7 Ellipsometry from Metal Surfaces ............................... 64
3.8 Resonances ni Small Metal Spheres ................................... 84
3.9 The Electric Fields Near the Surface Inside the Metal ................. 25
3.10 The Photoemission dleiY from Metal Surfaces ...................... 55
3.11 Different Additional Boundary Conditions ............................ 85
4. Theoretical Concepts dna Models of Metal Surface Response ................ 26
4.1 Additional Boundary Conditions ro Susceptibility ...................... 26
4.2 Green's Functions for na Extended Hydrodynamic Model .............. 56
4.3 The Specular Reflection Model ..................................... 07
4.4 Microscopic Response Theory ......................................... 37
4.5 Collective dna Single-Particle Response ni the SCIB Model ............. 97
VII
5. Description of Nonlocal Effects by the Surface Response Functions
dJw) dna dll(w) ................................................. 98
5.1 Economical Presentation of Experimental Results: dL(w) , d\,(W) ...... 98
5.2 Boundary Conditions for the Asymptotic Fields ................... 19
5.3 The Long-Wavelength Limit ......................................... 69
5.4 Local Model for the Surface Region ............................. 201
5.5 Nonlocal Layer Model ni the Hydrodynamic Approximation ............... 701
5.5.1 No Surface Layer ........................................... 011
5.5.2 Local Limits ..........................................
5.5.3 gnoL Wavelength Limit ....................................... 311
5.6 Surface Plasmons ................................................... 611
5.6.1 Feibelman's Treatment ........................................ 611
5.6.2 Additional Surface Plasmon Modes ........................ 811
5.6.3 Experimental Evidence for Multipole Surface Plasmons .......... 021
7.5 F&urn6 ........................................................... 321
References ,,.,.,......,.........,,.,......,.................,.....~,... 521
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
VII
211
1. Introduction
In recent years optical methods such as differential reflection spectroscopy and
ellipsometry, electroreflection, excitation of surface plasmons, ultraviolet photo-
emission, etc., have extensively been applied to characterize clean and adsorbate-
covered metal surfaces both in vacuum dna in contact with an electrolyte. In the
near future, optical methods at ultraviolet frequencies probably will become even
more important, since synchrotron radiation as a convenient high intensity radia-
tion is becoming available in many places. Especially for the investigation of
metal/electrolyte interfaces optical methods are indispensable, since the power-
ful particle maeb methods of surface physics require vacuum conditions and are not
applicable at a liquid interface.
The optical surface effects ew discuss in this article occur in principle for
all frequencies of the incident light and are closely related to the perturbation
of the surface charge distribution, which is always induced by p-polarized light
(i.e. light with a finite normal component of the electric field vector) at the
interface between media of different bulk dielectric constants. At a metal surface,
these induced screening charges are confined to a narrow but finite surface region,
typically a few Angstroms thick or less. In principle, a modification of the charge
distribution at the surface, for instance by ad-atoms or an applied static elec-
tric field, affects the distribution of optically induced charges and thus the
surface electromagnetic fields dna the response properties. ehT optical techniques
mentioned in the beginning are os sensitive, that already minute modifications of
the surface constitution, for instance by much less than a monolayer of ad-atoms,
can eb clearly detected. In many cases, for instance for metal electrodes in elec-
trolytic cells, these optical techniques provide the most detailed information
available.
In this article ew will not consider inelastic light scattering which yam occur
at certain frequencies owing to transitions between particular collective modes or
single electron states of the metal or an absorbate.
The aim of optical experiments is to obtain information about the constitution
of the surface, e.g. the electronic charge distribution. Clearly, a reliable theo-
retical basis for the interpretation of such optical surface data is highly desir-
able. Unfortunately, classical Fresnel optics, which sah been applied successfully
to many problems of metal optics over many decades, is not sufficient for na ade-
quate understanding of many experimental results no a variety of metal surfaces.
In .pahC 3 of this article ew discuss several experimental examples which, in the
frequency region near the amsalp frequency (e.g.O.5~p ~ ~ 1.5~p), clearly refute
the predictions of classical optics.
ehT reason for eht failure of classical optics in this context is easy to under-
stand. ehT standard erudecorp of classical Fresnel optics /1.1/ is desab no the
noitpmussa that a yradnuob neewteb aidem nac eb detamixorppa yb a plane neewteb owt
suoenegomoh regions of different dielectric properties dna that the general solu-
tions of Maxwell's equations no both sides of eht interface are transverse diver-
gence-free electromagnetic waves, which are dehctam yb the standard yradnuob con-
ditions. roF p polarization, ilE dna (cid:127)D eht tangential tnenopmoc of the electric
field dna the lamron tnenopmoc of eht displacement field, respectively, must eb
continuous; therefore eht discontinuity of zE dna thus the induced surface egrahc
is proportional to the discontinuity of eht inverse dielectric constant I/~(m)
across the interface. In order to simulate within this hcaorppa the effect of a
slightly modified surface, for instance yb smota-da or na applied electric field,
eno sah to emussa a thin surface layer (thickness of the order 1 )X with a modified
dielectric constant ~s(m). Near eht amsalp frequency of this surface layer, I/Es(m)
semoceb large dna the classical Fresnel theory predicts a large induced charge dna
a large optical response, which sah on ecnednopserroc in the experimental findings.
In real systems there is on perfect screening dna on singular surface charge -ned
sity. ehT induced segrahc are spread over the distance of a screening length, typ-
ically of the order of I X in metals, dna eht induced segrahc in a thin surface
layer era coupled to those in the substrate. sA a ,ecneuqesnoc at the amsalp fre-
ycneuq of eht layer material on resonant amsalp oscillations era found in a very
thin adlayer.
ecniS the spatial extension of optically decudni charges is important for na
gnidnatsrednu of difference-spectroscopy experiments near the amsalp frequency, a
microscopic theory of electromagnetic surface esnopser smees a suitable basis for
the interpretation of hcus experiments. sseldeeN to say, that a full mutnauq -nahcem
ical theory, including lattice dna band-structure effects dna treating surface dna
bulk esnopser properties no equal footing, would require a suodnemert laciremun
effort dna sah ton even neeb attempted. ehT first ambitious work towards this direc-
tion sah neeb presented dna recently reviewed yb NAMLEBIEF /1.2/. eH deredisnoc
free-electron metals within the jellium model, which replaces the metal ions yb a
suoenegomoh structureless dnuorgkcab of positive charge, thereby neglecting lattice
effects, demussa a realistic effective surface potential, which speek eht electrons
inside the metal, dna calculated the optical response within the random phase ap-
proximation (RPA). ehT results of Feibelman's microscopic theory are in remarkably
good agreement with the measured photoyield spectrum of aluminum in a broad fre-
quency interval including the plasma frequency (cf. Sect. 3.10). ehT microscopic
calculation is very involved dna dependent no massy numerical work. These complica-
tions have prohibited its ready application to a variety of optical problems even
for simple metals.
ehT overwhelming majority of optical surface data is, however, not no free-elec-
tron metals but no noble metals dna transition metals, where interband transitions
are nwonk to eb very important for the optical response properties. Then neither
the classical Fresnel optics nor the APR calculations based no the jellium ledom
provide a suitable basis for the evaluation of experimental data: ehT former cannot
describe the spatial spread of induced charges, although the correct bulk dielectric
constant nac eb taken as input. ehT latter describes the charge distribution near
the surface well, but cannot account for the correct bulk response.
ehT main purpose of this article is to present a detailed discussion of the
merits, but also of the limitations, of a simple phenomenological generalization of
classical optics, which nac describe both, the spatial spread of induced charges
dna the correct bulk dielectric properties. ehT idea of this so-called "hydrodynamic
approximation" (HD) is to include gnoma the fundamental solutions of Maxwell's equa-
tions, which build pu the electromagnetic field in the metal, not only the diver-
gence-free transverse waves of Fresnel optics but also the longitudinal wave solu-
tions, which occur if the spatial dispersion, i.e. the k dependence in Fourier-
space, of the longitudinal dielectric function ~(k,m) is taken into account. With
the model assumption
~(k,m) = ~b(~) - ~/ ~(~+iT) - kB 2 (1.1)
of the DH dna tm = EL(O,m) for the transverse dielectric function, the longitudinal
eigenmodes of Maxwell's equations are plasma waves dna the optically induced, spati-
ally spread charge density, the divergence of the longitudinal electric field, is
approximated as plasma oscillations of the conduction electrons. ehT material para-
meters in (1.1) account for the correct bulk response properties. Just as the clas-
sical Fresnel optics, the DH works with matching conditions for the fields at na
interface between two media with different, but no both sides spatially constant
material parameters. Since the amplitudes of transverse dna longitudinal waves must
eb determined, besides the standard matching conditions "additional boundary con-
ditions" (ABC) are formulated.
In .pahC 2 ew explain the hydrodynamic approximation in detail dna in .pahC 3 ew
present a large number of different experimental examples which could eb understood
dna interpreted within this approach, whereas classical Fresnel optics failed. eW
want to emphasize that this success is not accidental dna that the inclusion of
plasma waves in metal optics is not just a theoretical trick. nO the contrary, there
is experimental evidence for the optical excitation of plasma waves in metals dna
ample justification from microscopic models.
Historically, plasma waves were predicted theoretically by MHOB dna SENIP /1.3/
in the fifties as eigenmodes in a suoenegomoh metal. Subsequently they were studied
by "longitudinal probes", i.e. by fast electrons transmitted through metal foils,
which loose energy of quanta pm~ no excitation of plasmons /1.4/. In 1964 RETUAS
/1.5/ pointed out that these longitudinal plasma waves should eb included in metal
optics, since they are solutions of Maxwell's equations if spatial dispersion, i.e.
non-locality of the longitudinal electronic response is taken into account. Sauter
required as CBA the continuity of the normal component of the current density and
published in 1967 a calculation /1.6/ of reflection from a metal surface, together
with a microscopic justification given by NNAMTSROF /1.7/ using Boltzmann's equa-
tion. nO the basis of essentially the emas macroscopic approach, KYNLEM and NOSIRRAH
/1.8/ predicted a few months later the possibility of resonant excitation of plas-
snom in thin metal films by electromagnetic waves. Such resonances were observed
experimentally in 1970/71 /1.9, i0/, dna were the first direct proof that plasma
waves are indeed excited optically.
Of course, the hydrodynamic approximation is a phenomenological approach and sah
its limitations. Collective plasma oscillations are only one possible response edom
of the electron system. Optical excitation of individual electron-hole pairs is
also possible near a surface dna provides an additional response and absorption
mechanism absent in the free electron model of bulk metals. Such effects are not
included automatically in the ,DH but they can eb simulated by a suitable choice of
parameters, e.g. for the damping in a surface layer. In Chap. 4 ew discuss the lim-
itations of the DH and ew review other phenomenological dna microscopic approaches
to metal optics. From the comparison of different approaches ew get the impression,
that the DH is easiest to handle and, eventually with a suitably chosen surface
layer, flexible enough to provide a good, at least qualitative, understanding of
surface response properties. It is interesting to note that the physical content
of Feibelman's microscopic result for the photoyield of aluminum saw explained by
model calculations within the ,DH sa will eb discussed in Sect. 3.10.
In the final Chap. 5 ew study the surface response functions d(cid:127) dna dll(m),
which were introduced in this form by NAMLEBIEF /1.2/ dna are won considered by
ynam authors /i. II/. These surface response functions contain the full information
about the nonlocal surface response dna determine the macroscopic quantities, which
can eb measured in different experimental situations. They provide a useful frame
for the presentation of experimental results, without imposing restrictive model
assumptions, dna are convenient for the comparison of experiment and theory. oT
clarify the physical implications of the peculiar frequency dependence of the sur-