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NON-LOCAL ELECTROMAGNETIC EFFECTS AT METAL SURFACES
S. Lundqvist, P. Apell
To cite this version:
S. Lundqvist, P. Apell. NON-LOCAL ELECTROMAGNETIC EFFECTS AT METAL SURFACES.
Journal de Physique Colloques, 1983, 44 (C10), pp.C10-305-C10-314. �10.1051/jphyscol:19831061�.
�jpa-00223519�
JOURNAL DE PHYSIQUE
Colloque CIO, supplément au n°12, Tome M, décembre 1983 page C10-305
NON-LOCAL ELECTROMAGNETIC EFFECTS AT METAL SURFACES
S. Lundqvist and P. Apell
Institute fop Theoretical Physios, Chalmers University of Technology, S-412 96 GOteborg, Sweden
Résumé - on discute de la réponse non-locale d'une surface métallique irradiée en termes de densité induite en réponse à un champ externe.Le champ et la charge induite au voisinage de la surface peuvent être déterminés en résolvant les équations de Maxwell dans la région de surface et en ajustant la solution aux ondes incidente et réfléchie dans le vide et à la solution asymptotique dans le volume du solide.
Pour les champs incidents dont les longueurs d'onde sont grandes par rapport à l'étendue de la région.de surface,la grandeur fondamentale est le centre de gravité de la charge d'écran.Cette fonction complexe de la fréquence détermine de nombreuses propriétés telles que la réflectivité,la dispersion des plasmons de surface,les potentiels images,l'interaction de Van der Waals entre une molécule et une surface métallique,etc..
Des considérations analogues à celles présentées plus haut pour une surface plane ont été aussi appliquées â l'étude des
propriétés de surface d'une petite particule sphérique.
Abstract - The non-local response of an irradiated metal surface is discussed in terms of the induced density in response to an external field.The field and the induced charge near the sur- face can be determined by solving Maxwell equations in the sur- face region and match the solution to the incoming and reflec- ted wave in vacuum and to the asymptotic solution in the bulk of the solid.
For incoming fields of wave lengths long in comparison with the extension of the surface region a key quantity is the center of gravity of the screening charge.This complex function of the frequency determines a number of properties such as the reflec- tivity,the surface plasmon dispersion,image fields,the Van der Waals interaction between a molecule and a metal surface,the damping of an excited molecule,etc.Considerations similar to the ones for a planar surface have also been applied to the surface properties of a small metallic sphere.
I - INTRODUCTION
The study of the electromagnetic fields and dynamical screening in the surface region of metals is of key importance for our understanding of a variety of physical phenomena.The non-local effects near the sur- face will influence the scattering and absorption of light and will be of importance for the description of properties like the reflec- tivity, Raman scattering,luminescence and second order harmonic gene- ration.They are also important for the interaction between particles and the metal surface.Examples of such effects are the image fields,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831061
JOURNAL DE PHYSIQUE
the Van der Waals interaction and the damping of an excited molecule outside a metal surface.
Several different theoretical models have been used to study the dy- namical screening effects.The hydrodynamical approach was introduced by Sauter /l/,who pointed out that the longitudinal bulk plasmons should be included in metal optics.The hydrodynamical model has been extensively used by many authors and we would particularly mention the work by Forstmann and coworkers / 2 / .
A step beyond the hydrodynamical approach is the Semiclassical model.
We would like to mention here the classical papers by Kliewer and Fuchs / 3 , 4 / and by Flores and Garcia-Ploliner /5,6/.
In order to obtain a satisfactory description of the dynamical scree- ning in the surface region one has to use a quantum mechanical theory.
Models like the RPA often simplified by specific assumptions about the surface profile have been discussed by many authors.We would in particular mention the work by Feibelman,who used a self-consistent approach based on the wave functions and density profile for jellium calculated by Lang and Kohn /7/.We refer to a recent review by Feibelman /8/,which contains an extensive list of references to the literature in this field.
The hydrodynamical model is the simplest theoretical model to describe non-local electromagnetic effects.lhe external field induces a longi- tudinal charge density oscillation which gives rise to a longitudinal field in addition to the transverse field in the metal.The model is adequate in the frequency regime around the bulk plasma frequency, where such charge density oscillations,the plasmonstform the dominant
=ode of excitation,and the plasmons contribute directly to the opti- cal properties.Below the plasma frequency the plasmons are not excited and the excitation of individual electron-hole pairs is the dominant mode of excitation,but these excitations are not included in the hydrodynamical approach.
The semi-classical approach uses a Landau-Vlasov transport equation to describe the motion of the electronstand they are either specular- ly or diffusively scattered against the surface.The model gives a good description of the screeniny and the fields in the bulk,but not in the immediate vicinity of the surface where interference effects, which are neglected in the semi-classical theory,may be important and where the actual shape of the surface profile and the corresponding deviation from a plane wave state is important.The semi-classical model is therefore not particularly suitable to study surface-sensi- tive properties.
The quantum mechanical approach is in principle free from the limita- tions of the hydrodynamical or semi-classical models.The elementary modes of excitation are electron-hole excitations and plasmons in the presence of the surfaceland the wave-functions are calculated in a self-consistent way together with the surface barrier potential.
Calculations using different assumptions about surface profiles and wave functions show that the dynamical screening effects depend very sensitively on the choice of surface profile and wave function.The numerical values between different models can easily differ by an or- der of magnitude.This indicates that most of the model calculations in the literature have at most some qualitative significance and that realistic calculations,where wave functions and potentials are calcu- lated self-consistently,are needed for quantitative results.
The study of non-local electromagnetic effects has up to now been li- mited to the idealized model of jellium with a planar surface-Some theoretical results have also been obtained for a small spherical jellium particle.Tne extension of the theory to the case of a planar periodic surface would seem rather straightforward.The case of rough surfaces,e.g. surfaces with gratings,bumps or pits,present a real challenge.The treatment of non-local effects for such systems is still in a very preliminary stage.The case of aggregates of small metallic
p a r t i c l e s i s a n o t h e r example where rruch work h a s been done u s i n g l o c a l t h e o r i e s b u t where t h e n o n - l o c a l p r o p e r t i e s have n o t y e t been s t u d i e d .
I1
-
B a s i c s o f non-local p r o p e r t i e s .The c l a s s i c a l t h e o r y o f t h e d i e l e c t r i c p r o p e r t i e s c h a r a c t e r i z e s a f i - n i t e s o l i d by i t s d i e l e c t r i c c o n s t a n t which changes d i s c o n t i n u o u s l y a t t h e s h a r p s u r f a c e .
For t h e i n t e r f a c e between an i s o t r o p i c s o l i d and vacuum t h e d i e l e c t r i c f u n c t i o n jumps from t h e v a l u e ~ ( w ) i n t h e s o l i d t o u n i t y i n vacuum when we p a s s t h e i n t e r f a c e . T h e e l e c t r o m a g n e t i c f i e l d s a r e determined everywhere by s o l v i n g Maxwell e q u a t i o n s and u s i n g t h e wellknown match- i n g c o n d i t i o n s a t t h e s u r f a c e r e q u i r i n g simply t h a t t h e t a n g e n t i a l components o f t h e e l e c t r i c and magnetic f i e l d s a r e c o n t i n u o u s a c r o s s t h e i n t e r f a c e .
For a p l a n a r s u r f a c e and i n c i d e n t s - p o l a r i z e d l i g h t , t h e c l a s s i c a l t h e o r y g i v e s a n e s s e n t i a l l y c o r r e c t d e s c r i p t i o n o f t h e o p t i c a l pro- p e r t i e s . T h e r e a s o n i s t h a t t h e normal component o f t h e e l e c t r i c f i e l d v a n i s h e s everywhere and t h e r e f o r e t h e r e w i l l be no induced c h a r g e a t t h e i n t e r f a c e .
For p - p o l a r i z e d l i g h t on t h e o t h e r hand we have a non-vanishing nor- mal component of t h e i n c i d e n t e l e c t r i c f i e l d , w h i c h w i l l have a n a - b r u p t change a c r o s s t h e i n t e r f a c e . A s a r e s u l t we s h a l l have a s i n g u l a r
6 - f u n c t i o n induced c h a r g e a t t h e i n t e r f a c e - I t i s c l e a r t h a t t h i s s i n g u l a r c h a r g e w i l l n o t be p r e s e n t i n a m i c r o s c o p i c t h e o r y and t h a t t h e f i e l d s c a l c u l a t e d a c c o r d i n g t o t h e classcial t h e o r y w i l l be un- p h y s i c a l i n t h e s u r f a c e r e g i o n and do n o t g i v e a u s e f u l d e s c r i p t i o n of t h e dynamical s u r f a c e s c r e e n i n g ,
I n o r d e r t o t r e a t t h e s u r f a c e s c r e e n i n g one needs a p r o p e r micro- s c o p i c model o f t h e s u r f a c e and t h e e l e c t r o n i c e x c i t a t i o n s n e a r t h e s u r f a c e . T n e s h a r p i n t e r f a c e h a s t o b e r e p l a c e d by a c o n t i n u o u s s u r - f a c e p r o f i l e where t h e e l e c t r o n d e n s i t y i n c r e a s e s smoothly from z e r o and r e a c h e s t h e c o n s t a n t b u l k v a l u e ( f o r j e l l i w n ) i n s i d e t h e m e t a l . The i n c i d e n t l i g h t w i l l c o u p l e v e r y s t r o n g l y t o t h e e l e c t r o n s i n t h e s u r f a c e r e g i o n - T h e p r e s e n c e o f t h e s u r f a c e b r e a k s t h e t r a n s l a t i o n symmetry normal t o t h e s u r f a c e ( t h e z - d i r e c t i o n ) and makes i t p o s s i b l e t o e x c i t e e l e c t r o n - h o l e p a i r s o f s h o r t wavelengths a l l t h e way down t o wavelengths o f t h e o r d e r of t h e Fermi momentum.The i n c i d e n t l i g h t i n - d u c e s s t r o n g e l e c t r o n - h o l e p a i r e x c i t a t i o n s i n t h e s u r f a c e r e g i o n , b o t h v i r t u a l e x c i t a t i o n s which c o r r e s p o n d t o p e r i o d i c f l u c t u a t i o n s i n t h e s u r f a c e c h a r g e d e n s i t y w i t h t h e f r e q u e n c y o f t h e i n c i d e n t l i g h t , a s w e l l a s r e a l e x c i t a t i o n s o f e l e c t r o n - h o l e p a i r s c o r r e s p o n d i n g t o t h e d i s s i p a t i o n o f e l e c t r o m a g n e t i c e n e r g y by e x c i t i n g e l e c t r o n - h o l e p a i r s .
I f t h e f r e q u e n c y i s a r o u n d o r above t h e b u l k plasmon frequency,we s h a l l a l s o have d e n s i t y o s c i l l a t i o n s c o r r e s p o n d i n g t o b u l k plasmons.
The mechanism j u s t d e s c r i b e d means t h a t t h e i n c i d e n t l i g h t i n t e r a c t s w i t h e l e c t r o n s and h o l e s t w h i c h a r e d e s c r i b e d by wave f u n c t i o n s having a n a p p r e c i a b l e r a n g e i n s p a c e . T h i s i m p l i e s t h a t t h e d i e l e c t r i c pro- p e r t i e s c a n n o t be d e s c r i b e d i n c l a s s i c a l terms.The d i e l e c t r i c r e s p o n s e w i l l be a n o n - l o c a l f u n c t i o n ~ ( r , r - , w ) , w h i c h f o r v i s i b l e l i g h t and
j e l l i u m r e d u c e s t o E ( Z , Z ; W ) . T ~ Z a i f f e r e n c e between d i f f e r e n t forms o f l o c a l and n o n - l o c a l d i e l e c t r i c f u n c t i o n s i s i l l u s t r a t e d i n f i g u r e 1 To c h a r a c t e r i z e t h e d i f f e r e n c e between t h e c l a s s i c a l t h e o r y and a m i c r o s c o p i c d e s c r i p t i o n we emphasize a g a i n t h e main p o i n t s :
One h a s t o i n t r o d u c e t h e m i c r o s c o p i c d e n s i t y p r o f i l e r a t h e r t h a n t h e s t e p p r o f i l e i n c l a s s i c a l t h e o r y .
The d i e l e c t r i c r e s p o n s e of t h e s u r f a c e i s n o n - l o c a l r a t h e r t h a n l o c a l .
The d i e l e c t r i c r e s p o n s e depends on t h e d e t a i l e d p r o p e r t i e s of t h e e l e c t r o n - h o l e e x c i t a t i o n s and t h e b u l k plasmons.
JOURNAL DE PHYSIQUE
Fig. 1
-
Different forms of local and non-local dielectric functions.One characteristic feature of the n o n - l o c a l response is that the normal component of the electric field will set up charge oscilla- tions in the metal which will produce a longitudinal field inside the meta1,as illustrated in figure 2.
The method to determine the fields is discussed in /9/.0ne has to
~ntegrate Maxwell equations over the surface region and match the solution to the incoming and reflected waves in vacuum and to the bulk solutions inside the surface region.Because of the non-locality effects the surface region will extend deeper than the density pro- file region-Whereas the density profile essentially reaches the bulk value within a few A the surface region may extend considerably deep- er into the meta1,but still a small distance compared with the skin depth.This is illustrated schematically in figure 3,and the procedure is indicated in figure 4.
A key difference between the microscopic description and the classical theory is that the unphysical singular charge at the interface is now replaced by a smooth induced charge distribution as indicated by the dashed curve in figure 4.This oscillating induced charge density is
Fig. 2
-
Incident,reflected and transmitted waves for p-polarized incident light.density profile region
I vacuum
I surface region!
Fig. 3
-
The variation of the normal component of the electric field, which determines the width of the surface region.the generalization to finite frequencies of the static induced elec- tron density calculated for jellium by Lang and Kohn /lo/.
The induced electron density Gp(z,w) is a very convenient quantity to discuss the surface properties.We note that 6p(z,w)is complex for w
+
0.It is characteristic for both the static and dynamic induced densities that they peak strongly in the neighbourhood of the jellium edge-In the static case Lang and Kohn defined the location z of the equiva- lent image plane through the formula 0
Fig. 4
-
The three regions with matching at zl and z .The dashed curve indicates the result for the induced charge density.$he matching con- ditions determine the reflectivity p.
P
CIO-310 JOURNAL DE PHYSIQUE
The first moment of Gp(z,w) is a very convenient quantity to discuss dynamical screening effects and one defines a length
For w 0 d,(w) is a complex function. Red,(w) determines the posi- tion of the equivalent image-plane as a function of w according to the formula
where E (w) is the classical (Drude) dielectric function. Imd,(w) is the absorptive part.The absorption is due to the real excitation of electron-hole pairs and to plasmonstif w > wpl.
It is clear that the induced charge density Gp(ztw) corresponds to the retarded response of the surface and the same is true for d,(w).
The function d,(w) therefore fulfills the Kramers-Kronig relations Imdl (w'f
Red, (61) = 2.~dUc, d L - w L Imd, (w) =
-2.
n,-Zd
Redl (6)0 WCL
-
a-Persson and Ape11 /11/ have used these relations to derive several useful sum rules-The most important are the following ones
Here
n ( r , )
is a function only of the electron gas density parameter r ,and is given numerically in their paper.I11
-
Power absorption and the reflectivity.We demonstrated in the preceding section that the dielectric proper- ties of the interface change in a continuous way and that the non- locality of the dielectric response must be taken into account.We now calculate the absorption of light which also gives the result for the reflectivity pp.
The power absorption a(w) is given by the time-averaged power absorb- ed in the metal normalized to the incident flux I =c/8~r*cosQ*A,A be- ing the irradiated area and 0 the angle of incidence. 0
It is convenient to split a ( w ) into two parts a(w)=al (w)+a2 (w) as follows
a (w) = -w/8111 * I ~ J ~ ~ x ( D ~ * E , ~
+
D:.E, cl )1 0 ( 7 )
and
a2 (w) = -w/8n1~.1mJd~x :D (EL-EY1)
where we have added and subtracted the perpendicular field component EZ1 for a sharp interface which has a step at z=0 connected to the
singular classical surface charge.
We notice that al(w) is a bulk-like contribution while a2(w) is by construction a surface property since it invokes the difference bet- ween the actual field perpendicular to the surface and the field for the sharp interface.a2(w)zO for a sharp surface.
Use of the boundary conditions on the tangential components of the electric and magnetic fields and assuming that E I I t D , I and D, have the same
form
as in classical theory gives the formulas /12/al (w) = COSO 11-p
1
' ~ e (q€/pt)P ( 9 )
and
2 1-E
a (w) = cos811-p I21m(qsin e ~ , ( w ) ) / l z 1 ~
2 P (10
Here q=w/c is the wave number of the incident light and pt is the z -
component for the transmitted wave.~=~(w) is the classical (Drude) dielectric function.2 is the surface impedan~e~related to the reflec- tivity p through the formula
P
We can also express the power absorption a(w) directly in terms of the surface impedance Z as follows:
Comparing this formula with equations (9) and (10) we find a surface contribution proportional to d,(w) :
The classical surface impedance z ~ ~ = ~is the result for a sharp ~ / ~ E
interface,where by definition dl(w)vanishes.
The term sinz% is essential and shows how important the perpendicular field component is in seeing any effect from the surface region.
We have shown that the optical properties of the vacuum-metal inter- face are completely determined by our formulas for pp and Z and that the surface contribution is characterized by the function d,(w),intro- duced in section II.The arguments are elementary; we calculated the Joule heating together with the asymptotic forms of the fields outside and inside the medium.No additional boundary conditians was used; only the continuity of the tangential components of the electric and mag- netic fields was used.
The result for the surface impedance Z is good to first order in the ratio between the width of the surface region and the waye length of the incident radiation.
With some care,the surface plasmon dispersion can be found from the formula for the reflectivity.?', surface wave would correspond to the situation that we have a reflected waye without an incident wave, which formally corresponds to p +a .En the non-retarded limit one obtains in the limit of small kP(k<<kF)
w2 = W; (l+kd,) (14)
C10-312 JOURNAL DE PHYSIQUE
where wg=w / J 2 and k is the wave vector in the plane of the surface.
This resulg was first obtained by Harris and Griffin /13/ using the RPA and soon thereafter derived using electromagnetic theory by Garcia -Moliner and Flores /14/.
IV
-
Brief remarks about other applications.We introduced in section I1 fhe complex length d,(w) as a convenient quantity characterizing the electron density induced by the incident light wave.The real part gives the centre of gravity of the induced charge and serves to define the dynamical image planelwhereas the imaginary part describes the absorption by real excitations of elec- tron-hole pairs or plasmons.ln the last section we found that the surface contribution to the reflectivity or surface impedance is de- termined by d,(w).This is also true for a number of other properties where the coupling to the surface is weak and involves some distance from the surface.We shall mention very briefly some of these appli- cations in this section,
Let us first consider a molecule with polarizability a(wJ situated at some distance d outside the jellium edge and influenced by an ex- ternal field of frequency w.This field induces a periodic variation of the dipole moment 3 of the molecule.However,this motion of the dipole induces a charge density oscillation in the metal surface,which gives rise to an image field acting back on the dipole.The total field acting on the molecule is therefore the sum of the external and image fieldsrand we obtain the equations:
ext + Eimage
p = C L ( W ) . (
5 -
)where PL
image = E(w)-1-
EL E (w) +1 4 (d+dyp (wJ 1
and d ~ p (w) (defined in equation ( 3 ) ,is the dynamical image plane.We note that d r g ( w ) has a finite imaginary pare due to electron-hole excitations (and plasmons if w>wplJ.
The effect of the image field is to shlft the molecular dipolar fre- quencies and also to introduce a width to the levels Because of the creation of real excitations in the metal surface-The system one actually studies in such cases 2s the coupled system of the molecule and the induced polarization of the metal surface.
Closely related is the van der FJaals interaction between a molecule and a metal surface.In the classical theory the interaction between a molecule of polarizability a(w] at a distance d outside a metal having a dielectric function ~(w),is given by /15/
This result is modified if we have a smooth density profile and has to be replaced by the more general formula
where
2 co wInds(w) dIp(iu) = -.Jdw
71 O w L + u L
which shows the role of the dynamically induced electron density via the dynamic image plane.
We have already mentioned that dissipative effects due to the real excitation of electron-hole pairs (and possibly plasmons) modify the classical result.An experimentally well studied case is the damping
of an excited molecule some distance d (-1008) outside a metal sur- face.There are two major mechanisms contributing to the damping.One is the "bulkv-like damping coming from the damping terms in the classical dielectric function ~(w),which varies as dc3.yhe other is the surface damping due to the excitation of electron-hole pairs which varies as d-'.
We would finally like to mention that the approach discussed here has also been successfully applied to the optical properties of a small spherical jellium particle /16/.The surface profile seems to give a redshift of the spherical plasma frequency and it seems to give an appreciable enhancement of the field,particularly in the regime of the spherical plasma frequency.We believe that this model of spherical particles has some validity for radii of the order of 308 or larger.
For very small particles effects of quantization and electron gas pressure will of course be important.
V
-
Concluding remarks.The discussion given in this review has been limited to the case of planar jellium surfaces.For such surfaces many properties can be
discussed in terms of ds(w) as long as the wave length of the incident light is long in comparison with the width of the surface region.dr(w) is related to the density fluctuations perpendicular to the surface.
For non-jellium surfaces or even,say,jellium with adsorbates,one has also to consider induced motions in the plane of the surface and one would then have to introduce a new function dll(wJ,which measures the current fluctuations in the plane of the surface.Most of the results still hold but we have to replace d r ( ~ ) ,for instance by d ~ ( w ) -dll(@) in the surface plasmon dispersion relation in equation (14).
Most important to consider is also the effects introduced by strong effects of the periodicity and in particular the effects of excita- tions of electrons that are more localized in space like d or f electrons.
We have already mentioned in the introduction the case of non-planar surfaces.The study of models of rough surfaces ?s a real challenge for future research into the non-local properties of surfaces.
Acknowledgements
Much of the preparation of this review was done when one of the au- thors (S.L.) spent a memorable sabbatical at the Institute for Theoretical Physics and the Department of Physics at the University of California,Santa Barbara.He wishes to express his sincere grati- tude in particular to Drs W. Kohn and R. Schrieffer.Me also wish to express our sincere thanks to Dr E. Burstein for the many inspiring discussions we have had with him on these topics over the years.
Financial support from the Swedish Natural Science Research Council is gratefully acknowledged.
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C10-314 JOURNAL DE PHYSIQUE
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