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Experimental studies of optical surface excitations
SURFACE
SCIENCE 34 (1973) 50-61 0 North-Holland
EXPERIMENTAL
STUDIES
Publishing Co.
OF OPTICAL
SURFACE
EXCITATIONS
B. FISCHER Max-Planck-Institut
fiir Festkiirperforschung, Stuttgart I, Germany N. MARSCHALL
Physikalisches fnstitut der ~nive~sit~t FrunkfzirtlMai~,
Germany
H. J. QUEISSER Max-Planck-Institut
fiir Festktirperforschun,?, Stuttgart I, Germany
Experimental methods for optical excitation of nonradiative surface waves, such as surface plasmons and surface phonons, are reviewed. Results for surface plasmons in InSb obtained by using samples with line gratings in the surface and by using the technique of attenuated total reflection (ATR) are compared. Best agreement between the theoretical and experimental surface plasmon dispersion curve o(k) of InSb is found when using excitation by ATR, since the sample surface need not be disturbed in this case. Surface phonons have been excited optically by applying ATR to alkali halide crystals and to the polar semiconductor GaP. While agreement with the theoretical dispersion curve is extremely good for GaP and also for CaF2, there are systematic deviations to lower frequencies for NaF and other alkali halides. In polar semiconductors the free carrier plasmon couples to the LO-phonon when op w OLO. This behaviour has been investigated theoretically and experimentally for the surface modes. For an interesting application of the ATR-technique, one can use the fact that dispersion and linewidth are a direct measure of the complex dielectric function e(ro) in the frequency range of the optical surface waves.
1. Introduction Surface excitations are a natural consequence of the fact that real crystals have boundaries to an outer mediumr). We shall restrict ourselves here to optical surface waves, such as optical surface phonons and surface plasmons. The boundary conditions determine the physical properties of the surface excitations. Frequencies and dispersive behaviour of the surface modes are determined in the region of long wavelengths: (I) by the dielectric function E(W) which consists of the contributions of the corresponding bulk excitations, (2) by the geometry of the crystal, and (3) by the dielectric constant q of the outer medium. 50
OPTICAL
The geometry
SURFACE
also determines
51
EXCITATIONS
whether
the modes
are radiative
or non-
radiative, where nonradiative means that such modes cannot couple directly to photons. Our main interest here will be concerned with the optical surface crystal. Such crystals have only waves at the boundary of a “semi-infinite” nonradiative modes. imately given by2)
The dispersion
k (WI = (4c>
relation
for this geometry
(1)
v)lf 9
[&I (0) V/(&I (0) +
is approx-
where k is the wavevector of the surface wave, o its frequency, Ed the real part of the dielectric function and r] the dielectric constant of the adjacent medium. Eq. (1) has a real solution for k at frequencies when E, (0)s - 9. One can see also that the phase velocity o/k of the surface wave is always smaller than the velocity c of light in vacuum. Such modes are called nonradiative since they cannot be excited by photons directly. Eq. (1) describes their dispersion in the region of mixed eigenstates of photon and surface excitation (“retardation region”)z). The dispersion curve approaches an asymptotic frequency CO,for k large compared to w/c, but still small compared to the Brillouin zone boundary value (or in the case of free carriers, k small compared to the wave vector k, at the Fermi surface). This asymptotic frequency is determined by the condition El
((4
=
-
(2)
9.
Eq. (2) is the limit of eq. (1) for k+co. Eq. (2) also follows immediately from the boundary conditions if retardation is neglected by setting c= co.
InSb -
Theory o
1
2 Wave
Vector
0
Experiment
SkP
k
Fig. 1. Dispersion of surface plasmons in InSb. Experimental results from reflectivity measurements at grating surfaces for different grating spacings and various dopings. Theoretical curve calculated from eqs. (1) and (3) with EL(= Ed) = 15.68 and y = 0.03 wg.
52
B. FISCHER,
N. MARSCHALL
AND H. J.QUEWER
The dispersion relation of surface plasmons in InSb is shown in fig. 1 as a specific example for these nonradiative modes4). This curve has been calculated
from eq. (1) using E
(co) =
eL
+
fzLco~/( - co2-
iyw) ,
for the dielectric function. Here or, is the plasma frequency and y the damping parameter of the free carriers, and sL is the background dielectric constant of the lattice. The entire dispersion curve is confined to the nonradiative region o/k-cc of the k-w plane. We will now consider experimental possibilities for the excitation and measurement of such nonradiative modes. For the sake of simplicity in the following, we will consider as the outer medium a vacuum, so q = 1. 2. Experimental 2.1.
PERIODIC
SURFACE
methods
STRUCTURES
Nonradiative modes can only be excited at wave vectors k larger than the photon wave vector w/c. This condition can be fulfilled, e.g., by using the momentum transfer of electrons or other particles scattered by these surface excitations5-8). However, we will here consider only methods of optical excitation. One method to get larger k vectors is to use a periodic structure in the surface which couples to the incoming photonspll). If there is a line grating in the surface with grating spacing d, and the photons are impinging on the surface under an angle CYin a plane of incidence perpendicular to the grating, then the components k, of the wave vectors tangential to the surface are k,=(w/c)sincl+lrr.2x/d
with
m=O,+l,
+2,....
The geometrical construction of the k-vectors in a k-o plane is shown in fig. 2. Excitation occurs only if the light is polarized parallel to the plane of incidence. mechanically produced gratings have been In most experiments, usedJ,s-ll). However, acoustical surface waves have been shown also to act as dynamic gratings inducing the excitation of optical surface plasmons12). These methodsg-12) using periodic structures were first applied to metals, where the plasma wavelengths are in the ultraviolet spectral region. It has not yet been possible to produce grating constants of the same order of magnitude as these short wavelengths so that only a small part of the dispersion curve near the light-line has been accessible to experiment [see fig. 2 and eq. (4)]. These experiments, however, have been done with high
OPTICAL
accuracy,
SURFACE
53
EXCITATIONS
so that small zone gaps at the k-values
of the artificial
superlattice
can be observed and investigated109 12). We have extended this type of measurement to free carrier plasmas in heavily doped semiconductors with plasma frequencies in the infrared region. Line gratings with spacings d of the order of the plasma wavelengths make
k,,,=w/c~sina+m~21C/d. m=O, fl, f2....
J-ko-w/c
I b s 3B
ll
-1”
C L?Y!JD
-2
l2
-3
+3 m
/
Fig. 2. Principle of excitation of nonradiative surface waves with the aid of a periodic surface structure. The upper part shows a beam incident on the surface with the structure. The lower part gives the construction of the points of excitation in the k-w plane. The wave vectors km with negative mare projected onto the half-plane with positive k, since the dispersion relation w(k) is quadratic in k.
it possible to get excitation of surface waves over the entire retardation region. Relatively high wave vectors in the region where the dispersion curve has reached its asymptotic frequency o, are accessible. The reflectivity spectrum of an InSb sample having an inscribed line grating with d= 30 urn is shown in fig. 3. The dips are caused by surface plasmon excitation. A series of those spectra with various d, cc, and doping level yielded the experimental dispersion relation which is shown in fig. 14). Agreement with the theoretical curve is rather good. However, we will compare this result with a second curve obtained by the technique of attenuated total reflection (ATR) which is another method to reach larger wave vectors k. This method, which was introduced by Otto to excite surface plasmonsi3), will be described in the following section.
54
B. FISCHER,
N. MARSCHALL
AND
H. .I. QUEISSER
?r lO’~n/crn’
d =30pm
Wave
Number
Fig. 3. Solid line, infrared reflectivity, at room temperature and normal incidence, of InSb sample with 7 x 10IR electrons/cm3 having a 30 pm-grating inscribed. Arrows indicate the fitted resonance frequencies at orders m. Dashed line, results from smooth surface for comparison.
2.2. ATTENUATED
TOTAL
REFLECTION
TECHNIQUE
At the boundary of a dielectric medium with index of refraction nP > 1, light can be totally reflected. In the state of total reflection there is no transmitted wave but an evanescent wave propagating along the surface and decaying exponentially on both sides away from the surface. An important fact is that the k-vector, k,, of this evanescent wave is increased compared to the wave vector in vacuum: k, = nP (o/c)
sin a.
In the range of total reflection l/n,
< since < 1,
(6)
so that w/c c k, < nplc
(7)
is the range of available k-values. Let us now consider the case where a sample is brought close to a prism surface within a distance of the order of the decay length of the evanescent wave. This wave leaking out of the prism can couple to a surface wave in the sample if frequency and wave vector are matched according to the dispersion relation of the surface wave. The optical arrangement and the geometrical construction of the point of excitation in a k-w plane are shown
OPTICAL
in fig. 4. The excitation
SURFACE
is observed
55
EXCITATIONS
experimentally
in the spectrum
of the
internally reflected intensity which is lowered around the resonance frequency from its hundred per cent value. The wave vector can be varied by changing
the internal
angle E of incidence.
k5-k0.ngsin Cl
Wave
Vector
k
Fig. 4. Principle of excitation of nonradiative surface waves via attenuated total reflection (ATR). The evanescent wave of the prism can cause excitation at wave vectors ks between w/c and kmax = np x w/c, where np is the index of refraction of the prism.
For practical purposes we used prisms shaped as parallelepipeds which have the convenient property that incoming and outgoing beams are always parallel to each other. Therefore, a rotation of the prism for changing the angle CLof incidence does not necessitate a rotation of the whole detecting system. The arrangement can be used simply in the transmission part of a commercial spectrometer. For studies of surface plasmons in semiconductors and surface phonons we used prisms made from silicon, which is transparent over a wide range in the interesting spectral region in the infrared. The relevant index of refraction is n( Si) = 3.4 18 14). The ATR-technique has two main advantages over the grating technique: (I) The surface of the sample need not be destructively perturbed. Therefore second order interactions of the surface waves with the periodic structure such as zone gaps or increased damping are avoided. (2) The ATR-spectra
56
B. FISCHER,
N. MARSCHALL
AND H. J. QUEISSER
can be calculated exactly with a multilayer reflection formulai7). The evaluation of the resonance frequencies for grated surfaces is much more complicated and only approximate since for exact theory the grating profile must be known is). In contrast, the calculated ATR-spectra show that in the limit of weak coupling, i.e. when the gap between sample and prism is large enough, the frequencies of the reflectivity minima in the ATRspectra directly yield the dispersion curve given by eq. (1). Moreover, the linewidths of the spectra can be evaluated to get the damping of the surface wavesi3). 3. Experimental
results
3. I. SURFACE PLASMONS Some ATR-spectra with surface plasmon in fig. 515). The dispersion curve obtained
excitation in n-InSb from those spectra,
a = 6.Opm
In Sb 0
1
300
wP=126.5cm-l I
1
350
are shown which can
LOO
I
lim-8
Wove Number
Fig. 5.
Spectra of attenuated total reflection showing surface plasmon excitation in InSb. The air spacing between sample and prism was a = 6 Wm.
be seen in fig. 6, shows that agreement with theory is improved compared with the experimental curve in fig. 1. We ascribe this improvement to the fact that the prism method utilizes optically polished surfaces whereas the mechanical destruction of the grating surfaces may easily cause electronically depleted surface layers which influence the frequencies of the surface plas-
OPTICAL
mons4*13). Rather
SURFACE
57
EXCITATIONS
drastic effects of this kind have been observed
by Ander-
son, Alexander and Bell’s). Concerning our experimental curve in fig. 1, it is, in fact, possible to fit the experimental points by a dispersion curve which is calculated for a surface having a depletion layer of about I urn depth. The average carrier concentration in this layer must be assumed at least one order of magnitude lower than in the bulk. The value of 1 urn,
// “p__-7_-_J___-_--_________-_--
"S
/
/--’ -
Theory . - - -
Experiment
InSb
2
1 Wave
Fig. 6.
as deduced surprisingly
Vector
Dispersion of surface plasmons
3
k
kP
in InSb, obtained from ATR-spectra.
from a simple step model for the depletion, large for such heavily doped semiconductors
cm”). Further work and corroborating data are needed tion in the case of scribed line gratings.
however, appears (> 10’ * electrons/ to clarify this situa-
3.2. SURFACE PHONONS In 1970, Ruppin suggested applying the ATR-technique to ionic crystals in order to excite optical surface vibrationslg). In the long wavelength region, the surface eigenstate is a mixed type of a photon and a surface phonon. In analogy to the bulk polaritons these modes are called surface polaritons. Their dispersion starts at the point o = wTo, k = k,, = wTo/c with the slope do/dk
= c/Cl + (so - E,)-
‘1,
where oTo is the transverse optical phonon frequency and E,, and E, are the low and high frequency dielectric constants, respectively. The limiting frequency w, for large k is found by using s(O) = &cc+ (%I - s~)/[~
-
(“/wTO)2]
3
(8)
58
B.FISCHER,N.MARSCHALI. AND H.J.QUEISSER
in eq. (2) and isl) us =
WTO[(FO
+ ~)/(&cc +
?>I",
where again q= 1. Eq. (9) has a formal analogy to the Lyddane-Sachs-Teller relation for the bulk phonon frequencies. Experimental investigations of the surface polariton dispersion have been done on the polar semiconductor GaP, on several alkali halides, and on CdF, and CaF,20-22). The experimental dispersion curves are usually compared with the simple theory based on the Lorentz-function of eq. (8) modified to allow for a constant damping term. Agreement is extremely good in the cases of GaP and CaF,, but there are large systematic deviations for the alkali halides. Experimental limiting frequencies w, lie considerably below the value predicted by eq. (9). This deviation has been demonstrated by Bryksin, Gerbshtein and Mirlin for KBr, NaCl, NaF and LiFse). Our own results on NaF essentially confirm these results; actually, w, is slightly higher in our experimentsss). We now want to discuss possible explanations for these deviations. First, one can think about surface layers of hydroxides or other compounds being created at the surface by chemical reaction with the surrounding air. This sounds reasonable since the alkali halides are well known to be hygroscopic and therefore tend to form such surface layers. There are arguments against this effect as an explanation for the observed deviations. At low values of the wave vector k, where the penetration depth of the surface wave is of the order of the wavelength (z 10 urn), the surface wave should not be affected that strongly. Typical thicknesses of hydroxide layers are 3001000 Ae4). Experimental work should, however, be done to clarify this point, e.g. by working with surfaces which have been freshly cleaved in vacuum or in a material which does not react chemically with the sample. As another possibility, we want to discuss the influence of the frequencydependence of the damping parameter y. It is well known from the Reststrahlen-reflectivity measurements on alkali halides that their spectra cannot be fitted using a constant damping. A complicated function y(o) which includes the contributions of all decay channels for the phonon must be usedss). Most important are the two-phonon processes which cause this damping function to assume one or more rather sharp maxima at about which is for the alkali halides in the region of oLo or slightly wTO+OTA~ below it. The properties of the damping function influence the Reststrahlenreflectivity spectra by producing minima in the region of high reflectivity between WTo and oLo 26). Bryksin et al. took into account these anharmonic contributions to the phonon damping. They then calculated the surface polariton dispersion for NaCl, and got much better agreement with their experimental dataea).
59
OPTICALSURFACEEXCITATIONS
We obtained a similar result for our NaF data, as shown in fig. 7. However, we have the problem that the damping function y(w) for NaF is not known at present and therefore has to be approximated, for instance by a Lorentz-function. In spite of the obvious improvement of the fit to the experimental points by using such a y(o)-function, there exist considerable theoretical problems concerning the evaluation of dispersion and linewidths
Wave vector
k
Fig. 7, Dispersion of surface polaritons in NaF. The theoretical curves I and II have been calculated using Sauter’s boundary conditions (see footnote of ref. 2), curve I with a constant damping y = 0.08 aTO, curve II with an approximated damping function I;(W).
to get ct (u) and ~~(0) in the case of frequency-dependent damping. For constant damping the linewidths of the ATR-spectra are a direct measure for the surface mode damping, which is essentially equal to the damping y of the bulk mode4v20). In fact, the experimental linewidths include further contributions arising from finite beam divergence and “radiative damping” by the prismls). We will omit here a detailed discussion of the difficulties associated with the evaluation of true linewidths, because of the theoretical problems mentioned in the footnote of ref. 2. From the results of Bryksin et al. and our own calculations, we conclude that the good agreement in the cases of GaP and CaF, is essentially due to the fact that the frequencies of the surface modes are in the region where the damping at room temperature is nearly constant. 3.3 SURFACE PLASMON-PHONON COUPLING We now want to turn to the effect of plasmon-phonon
coupling which
60
B. FISCHER,
N. MARSCHALL
AND
H. J. QUEISSER
can be studied very nicely in polar semiconductors. By changing the doping one can shift the plasma frequency wp through the longitudinal optical phonon frequency wLo. The frequencies of the coupled modes can be obtained from Raman scattering and reflectivity measurements 27, 2s). Theoretical considerations show that the optical surface waves should also be coupled. The coupled frequencies as function of carrier concentration have been calculated by Chiu and Quinn29). Wallis and Brion gave the dispersion relation for coupled surface modes30). Experimental work has been done by Genzel et al. on surface modes in small crystallites of Cd0 31), by Reshina, Gerbshtein and Mirlin on “semi-infinite” crystals of InSbsz), and by us on GaAsss). Reshina et al. obtained experimental points for nearly ten different coupling strengths. 4. Summary and conclusions We have considered experimental methods for optical excitation of nonradiative optical surface waves. The results presented are mainly concerned with the dispersion w(k) of these modes. In the long wavelength region dispersion and lifetime of the surface waves are determined essentially by the bulk parameters E,(U) and ~~(0). These parameters can be obtained from frequencies and linewidths of ATR-spectra; however, the evaluation of the linewidths is rather complicated, especially when the damping y is a function of frequency. A strongly frequency-dependent function y(w) influences also the dispersion relation as has been demonstrated for some alkali halides. Besides these bulk parameters, the influence of typical surface phenomena such as surface layers can be studied. Furthermore, the surface modes show also coupling between different types of excitations, plasmon-phonon and magnon-phonon coupling.
for instance
Acknowledgements We thank Professors R. J. Bell, H. Bilz, L. Genzel and J. J. Quinn for many interesting discussions. Financial support of the Fraunhofer Gesellschaft is gratefully acknowledged. References 1) R. Ruppin and R. Englman, Rept. Progr. Phys. 33 (1970) 149. 2) The derivation of the dispersion relation from Maxwell’s continuity conditions leads to eq. (l), where instead of EI(w) the complex dielectric function E(W)= EI(w) + isz (w) appears. [See for instance: M. Cardona, Am. J. Phys. 39 (1971) 1277.1 Difficulties with the formula including the complex E(W) arise from the fact that Maxwell’s continuity conditions are not complete for conducting media, which was shown by Sauter [F. Sauter, 2. Physik 203 (1967) 4881. Eq. (I) is a good approximation when
OPTICAL
3) 4) 5) 6) 7) 8) 9) 10)
I I) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33)
SURFACE
EXCITATIONS
61
~2< ~1. The derivation of the exact expressions for dispersion and lifetime of the surface waves, based on Sauter’s boundary conditions, will be published elsewhere. R. Fuchs and K. L. Kliewer, Phys. Rev. B 3 (1971) 2270. N. Marschall, B. Fischer and H. J. Queisser, Phys. Rev. Letters 27 (1971) 95. H. Boersch, J. Geiger and W. Stickel, Z. Physik 212 (1968) 130. J. B. Chase and K. L. Kliewer, Phys. Rev. B 2 (1970) 4389. H. Ibach, Phys. Rev. Letters 24 (1970) 1461. H. Raether, in: Springer Tracts in Modern Physics, Vol. 38, Eds. G. Hohler et al. (Springer, Berlin, 1965) p. 84. Y. Y. Teng and E. A. Stern, Phys. Rev. Letters 19 (1967) 511. R. H. Ritchie, E. T. Arakawa, J. J. Cowan and R. N. Hamm, Phys. Rev. Letters 21 (1968) 1530. D. Beaglehole, Phys. Rev. Letters 22 (1969) 708. J. Schoenwald, E. Burstein and R. F. Wallis, Bull. Am. Phys. Sot. [II] 16 (1971) 1409. A. Otto, Z. Physik 216 (1968) 398. C. M. Randell and R. D. Rawcliffe, Appl. Opt. 6 (1967) 1889. Physikalisches Institut der Universitlt N. Marschall, unpublished dissertation, Frankfurt/Main, Germany, 1971. W. E. Anderson, R. W. Alexander and R. J. Bell, Phys. Rev. Letters 27 (1971) 1057. H. Wolter, in: Hundbuch der Physik, Vol. 24, Ed. S. Fliigge (Springer, Berlin, 1956. p. 461. J. Hogglund and F. Sellberg, J. Opt. Sot. Am. 56 (1966) 1031. R. Ruppin, Solid State Commun. 8 (1970) 1129. N. Marschall and B. Fischer, Phys. Rev. Letters 28 (1972) 811. V. V. Bryksin, Yu. M. Gerbshtein and D. N. Mirlin, Fiz. Tverd. Tela 13 (1971) 2125 [Engl. Transl. Soviet Phys.-Solid State 13 (1972) 17791. V. V. Bryksin, Yu. M. Gerbshtein and D. N. Mirlin, Fiz. Tverd. Tela 14 (1972) 543. B. Fischer, N. Marschall and R. J. Bell, to be published. R. J. Bell, private communication, based on X-ray data. H. Bilz, L. Genzel and H. Happ, Z. Physik 160 (1960) 535. A. Mitsuishi, Y. Yamada and H. Yoshinaga, J. Opt. Sot. Am. 52 (1962) 14. A. Mooradian and A. L. McWhorter, in: Light Scattering Spectra of Solids, Ed. G. B. Wright (Springer, New York, 1969) p. 297. C. G. Olson and D. W. Lynch, Phys. Rev. 177 (1969) 1231. K. W. Chiu and J. J. Quinn, Phys. Letters 35A (1971) 469. R. F. Wallis and J. J. Brion, Solid State Commun. 9 (1971) 2099. L. Genzel and T. P. Martin, Surface Sci. 34 (1973) 33. I. I. Reshina, Yu. M. Gerbshtein and D. N. Mirlin, Fiz. Tverd. Tela 14 (1972) 1280. N. Marshall and B. Fischer, to be published.
SCIENCE 34 (1973) 50-61 0 North-Holland
EXPERIMENTAL
STUDIES
Publishing Co.
OF OPTICAL
SURFACE
EXCITATIONS
B. FISCHER Max-Planck-Institut
fiir Festkiirperforschung, Stuttgart I, Germany N. MARSCHALL
Physikalisches fnstitut der ~nive~sit~t FrunkfzirtlMai~,
Germany
H. J. QUEISSER Max-Planck-Institut
fiir Festktirperforschun,?, Stuttgart I, Germany
Experimental methods for optical excitation of nonradiative surface waves, such as surface plasmons and surface phonons, are reviewed. Results for surface plasmons in InSb obtained by using samples with line gratings in the surface and by using the technique of attenuated total reflection (ATR) are compared. Best agreement between the theoretical and experimental surface plasmon dispersion curve o(k) of InSb is found when using excitation by ATR, since the sample surface need not be disturbed in this case. Surface phonons have been excited optically by applying ATR to alkali halide crystals and to the polar semiconductor GaP. While agreement with the theoretical dispersion curve is extremely good for GaP and also for CaF2, there are systematic deviations to lower frequencies for NaF and other alkali halides. In polar semiconductors the free carrier plasmon couples to the LO-phonon when op w OLO. This behaviour has been investigated theoretically and experimentally for the surface modes. For an interesting application of the ATR-technique, one can use the fact that dispersion and linewidth are a direct measure of the complex dielectric function e(ro) in the frequency range of the optical surface waves.
1. Introduction Surface excitations are a natural consequence of the fact that real crystals have boundaries to an outer mediumr). We shall restrict ourselves here to optical surface waves, such as optical surface phonons and surface plasmons. The boundary conditions determine the physical properties of the surface excitations. Frequencies and dispersive behaviour of the surface modes are determined in the region of long wavelengths: (I) by the dielectric function E(W) which consists of the contributions of the corresponding bulk excitations, (2) by the geometry of the crystal, and (3) by the dielectric constant q of the outer medium. 50
OPTICAL
The geometry
SURFACE
also determines
51
EXCITATIONS
whether
the modes
are radiative
or non-
radiative, where nonradiative means that such modes cannot couple directly to photons. Our main interest here will be concerned with the optical surface crystal. Such crystals have only waves at the boundary of a “semi-infinite” nonradiative modes. imately given by2)
The dispersion
k (WI = (4c>
relation
for this geometry
(1)
v)lf 9
[&I (0) V/(&I (0) +
is approx-
where k is the wavevector of the surface wave, o its frequency, Ed the real part of the dielectric function and r] the dielectric constant of the adjacent medium. Eq. (1) has a real solution for k at frequencies when E, (0)s - 9. One can see also that the phase velocity o/k of the surface wave is always smaller than the velocity c of light in vacuum. Such modes are called nonradiative since they cannot be excited by photons directly. Eq. (1) describes their dispersion in the region of mixed eigenstates of photon and surface excitation (“retardation region”)z). The dispersion curve approaches an asymptotic frequency CO,for k large compared to w/c, but still small compared to the Brillouin zone boundary value (or in the case of free carriers, k small compared to the wave vector k, at the Fermi surface). This asymptotic frequency is determined by the condition El
((4
=
-
(2)
9.
Eq. (2) is the limit of eq. (1) for k+co. Eq. (2) also follows immediately from the boundary conditions if retardation is neglected by setting c= co.
InSb -
Theory o
1
2 Wave
Vector
0
Experiment
SkP
k
Fig. 1. Dispersion of surface plasmons in InSb. Experimental results from reflectivity measurements at grating surfaces for different grating spacings and various dopings. Theoretical curve calculated from eqs. (1) and (3) with EL(= Ed) = 15.68 and y = 0.03 wg.
52
B. FISCHER,
N. MARSCHALL
AND H. J.QUEWER
The dispersion relation of surface plasmons in InSb is shown in fig. 1 as a specific example for these nonradiative modes4). This curve has been calculated
from eq. (1) using E
(co) =
eL
+
fzLco~/( - co2-
iyw) ,
for the dielectric function. Here or, is the plasma frequency and y the damping parameter of the free carriers, and sL is the background dielectric constant of the lattice. The entire dispersion curve is confined to the nonradiative region o/k-cc of the k-w plane. We will now consider experimental possibilities for the excitation and measurement of such nonradiative modes. For the sake of simplicity in the following, we will consider as the outer medium a vacuum, so q = 1. 2. Experimental 2.1.
PERIODIC
SURFACE
methods
STRUCTURES
Nonradiative modes can only be excited at wave vectors k larger than the photon wave vector w/c. This condition can be fulfilled, e.g., by using the momentum transfer of electrons or other particles scattered by these surface excitations5-8). However, we will here consider only methods of optical excitation. One method to get larger k vectors is to use a periodic structure in the surface which couples to the incoming photonspll). If there is a line grating in the surface with grating spacing d, and the photons are impinging on the surface under an angle CYin a plane of incidence perpendicular to the grating, then the components k, of the wave vectors tangential to the surface are k,=(w/c)sincl+lrr.2x/d
with
m=O,+l,
+2,....
The geometrical construction of the k-vectors in a k-o plane is shown in fig. 2. Excitation occurs only if the light is polarized parallel to the plane of incidence. mechanically produced gratings have been In most experiments, usedJ,s-ll). However, acoustical surface waves have been shown also to act as dynamic gratings inducing the excitation of optical surface plasmons12). These methodsg-12) using periodic structures were first applied to metals, where the plasma wavelengths are in the ultraviolet spectral region. It has not yet been possible to produce grating constants of the same order of magnitude as these short wavelengths so that only a small part of the dispersion curve near the light-line has been accessible to experiment [see fig. 2 and eq. (4)]. These experiments, however, have been done with high
OPTICAL
accuracy,
SURFACE
53
EXCITATIONS
so that small zone gaps at the k-values
of the artificial
superlattice
can be observed and investigated109 12). We have extended this type of measurement to free carrier plasmas in heavily doped semiconductors with plasma frequencies in the infrared region. Line gratings with spacings d of the order of the plasma wavelengths make
k,,,=w/c~sina+m~21C/d. m=O, fl, f2....
J-ko-w/c
I b s 3B
ll
-1”
C L?Y!JD
-2
l2
-3
+3 m
/
Fig. 2. Principle of excitation of nonradiative surface waves with the aid of a periodic surface structure. The upper part shows a beam incident on the surface with the structure. The lower part gives the construction of the points of excitation in the k-w plane. The wave vectors km with negative mare projected onto the half-plane with positive k, since the dispersion relation w(k) is quadratic in k.
it possible to get excitation of surface waves over the entire retardation region. Relatively high wave vectors in the region where the dispersion curve has reached its asymptotic frequency o, are accessible. The reflectivity spectrum of an InSb sample having an inscribed line grating with d= 30 urn is shown in fig. 3. The dips are caused by surface plasmon excitation. A series of those spectra with various d, cc, and doping level yielded the experimental dispersion relation which is shown in fig. 14). Agreement with the theoretical curve is rather good. However, we will compare this result with a second curve obtained by the technique of attenuated total reflection (ATR) which is another method to reach larger wave vectors k. This method, which was introduced by Otto to excite surface plasmonsi3), will be described in the following section.
54
B. FISCHER,
N. MARSCHALL
AND
H. .I. QUEISSER
?r lO’~n/crn’
d =30pm
Wave
Number
Fig. 3. Solid line, infrared reflectivity, at room temperature and normal incidence, of InSb sample with 7 x 10IR electrons/cm3 having a 30 pm-grating inscribed. Arrows indicate the fitted resonance frequencies at orders m. Dashed line, results from smooth surface for comparison.
2.2. ATTENUATED
TOTAL
REFLECTION
TECHNIQUE
At the boundary of a dielectric medium with index of refraction nP > 1, light can be totally reflected. In the state of total reflection there is no transmitted wave but an evanescent wave propagating along the surface and decaying exponentially on both sides away from the surface. An important fact is that the k-vector, k,, of this evanescent wave is increased compared to the wave vector in vacuum: k, = nP (o/c)
sin a.
In the range of total reflection l/n,
< since < 1,
(6)
so that w/c c k, < nplc
(7)
is the range of available k-values. Let us now consider the case where a sample is brought close to a prism surface within a distance of the order of the decay length of the evanescent wave. This wave leaking out of the prism can couple to a surface wave in the sample if frequency and wave vector are matched according to the dispersion relation of the surface wave. The optical arrangement and the geometrical construction of the point of excitation in a k-w plane are shown
OPTICAL
in fig. 4. The excitation
SURFACE
is observed
55
EXCITATIONS
experimentally
in the spectrum
of the
internally reflected intensity which is lowered around the resonance frequency from its hundred per cent value. The wave vector can be varied by changing
the internal
angle E of incidence.
k5-k0.ngsin Cl
Wave
Vector
k
Fig. 4. Principle of excitation of nonradiative surface waves via attenuated total reflection (ATR). The evanescent wave of the prism can cause excitation at wave vectors ks between w/c and kmax = np x w/c, where np is the index of refraction of the prism.
For practical purposes we used prisms shaped as parallelepipeds which have the convenient property that incoming and outgoing beams are always parallel to each other. Therefore, a rotation of the prism for changing the angle CLof incidence does not necessitate a rotation of the whole detecting system. The arrangement can be used simply in the transmission part of a commercial spectrometer. For studies of surface plasmons in semiconductors and surface phonons we used prisms made from silicon, which is transparent over a wide range in the interesting spectral region in the infrared. The relevant index of refraction is n( Si) = 3.4 18 14). The ATR-technique has two main advantages over the grating technique: (I) The surface of the sample need not be destructively perturbed. Therefore second order interactions of the surface waves with the periodic structure such as zone gaps or increased damping are avoided. (2) The ATR-spectra
56
B. FISCHER,
N. MARSCHALL
AND H. J. QUEISSER
can be calculated exactly with a multilayer reflection formulai7). The evaluation of the resonance frequencies for grated surfaces is much more complicated and only approximate since for exact theory the grating profile must be known is). In contrast, the calculated ATR-spectra show that in the limit of weak coupling, i.e. when the gap between sample and prism is large enough, the frequencies of the reflectivity minima in the ATRspectra directly yield the dispersion curve given by eq. (1). Moreover, the linewidths of the spectra can be evaluated to get the damping of the surface wavesi3). 3. Experimental
results
3. I. SURFACE PLASMONS Some ATR-spectra with surface plasmon in fig. 515). The dispersion curve obtained
excitation in n-InSb from those spectra,
a = 6.Opm
In Sb 0
1
300
wP=126.5cm-l I
1
350
are shown which can
LOO
I
lim-8
Wove Number
Fig. 5.
Spectra of attenuated total reflection showing surface plasmon excitation in InSb. The air spacing between sample and prism was a = 6 Wm.
be seen in fig. 6, shows that agreement with theory is improved compared with the experimental curve in fig. 1. We ascribe this improvement to the fact that the prism method utilizes optically polished surfaces whereas the mechanical destruction of the grating surfaces may easily cause electronically depleted surface layers which influence the frequencies of the surface plas-
OPTICAL
mons4*13). Rather
SURFACE
57
EXCITATIONS
drastic effects of this kind have been observed
by Ander-
son, Alexander and Bell’s). Concerning our experimental curve in fig. 1, it is, in fact, possible to fit the experimental points by a dispersion curve which is calculated for a surface having a depletion layer of about I urn depth. The average carrier concentration in this layer must be assumed at least one order of magnitude lower than in the bulk. The value of 1 urn,
// “p__-7_-_J___-_--_________-_--
"S
/
/--’ -
Theory . - - -
Experiment
InSb
2
1 Wave
Fig. 6.
as deduced surprisingly
Vector
Dispersion of surface plasmons
3
k
kP
in InSb, obtained from ATR-spectra.
from a simple step model for the depletion, large for such heavily doped semiconductors
cm”). Further work and corroborating data are needed tion in the case of scribed line gratings.
however, appears (> 10’ * electrons/ to clarify this situa-
3.2. SURFACE PHONONS In 1970, Ruppin suggested applying the ATR-technique to ionic crystals in order to excite optical surface vibrationslg). In the long wavelength region, the surface eigenstate is a mixed type of a photon and a surface phonon. In analogy to the bulk polaritons these modes are called surface polaritons. Their dispersion starts at the point o = wTo, k = k,, = wTo/c with the slope do/dk
= c/Cl + (so - E,)-
‘1,
where oTo is the transverse optical phonon frequency and E,, and E, are the low and high frequency dielectric constants, respectively. The limiting frequency w, for large k is found by using s(O) = &cc+ (%I - s~)/[~
-
(“/wTO)2]
3
(8)
58
B.FISCHER,N.MARSCHALI. AND H.J.QUEISSER
in eq. (2) and isl) us =
WTO[(FO
+ ~)/(&cc +
?>I",
where again q= 1. Eq. (9) has a formal analogy to the Lyddane-Sachs-Teller relation for the bulk phonon frequencies. Experimental investigations of the surface polariton dispersion have been done on the polar semiconductor GaP, on several alkali halides, and on CdF, and CaF,20-22). The experimental dispersion curves are usually compared with the simple theory based on the Lorentz-function of eq. (8) modified to allow for a constant damping term. Agreement is extremely good in the cases of GaP and CaF,, but there are large systematic deviations for the alkali halides. Experimental limiting frequencies w, lie considerably below the value predicted by eq. (9). This deviation has been demonstrated by Bryksin, Gerbshtein and Mirlin for KBr, NaCl, NaF and LiFse). Our own results on NaF essentially confirm these results; actually, w, is slightly higher in our experimentsss). We now want to discuss possible explanations for these deviations. First, one can think about surface layers of hydroxides or other compounds being created at the surface by chemical reaction with the surrounding air. This sounds reasonable since the alkali halides are well known to be hygroscopic and therefore tend to form such surface layers. There are arguments against this effect as an explanation for the observed deviations. At low values of the wave vector k, where the penetration depth of the surface wave is of the order of the wavelength (z 10 urn), the surface wave should not be affected that strongly. Typical thicknesses of hydroxide layers are 3001000 Ae4). Experimental work should, however, be done to clarify this point, e.g. by working with surfaces which have been freshly cleaved in vacuum or in a material which does not react chemically with the sample. As another possibility, we want to discuss the influence of the frequencydependence of the damping parameter y. It is well known from the Reststrahlen-reflectivity measurements on alkali halides that their spectra cannot be fitted using a constant damping. A complicated function y(o) which includes the contributions of all decay channels for the phonon must be usedss). Most important are the two-phonon processes which cause this damping function to assume one or more rather sharp maxima at about which is for the alkali halides in the region of oLo or slightly wTO+OTA~ below it. The properties of the damping function influence the Reststrahlenreflectivity spectra by producing minima in the region of high reflectivity between WTo and oLo 26). Bryksin et al. took into account these anharmonic contributions to the phonon damping. They then calculated the surface polariton dispersion for NaCl, and got much better agreement with their experimental dataea).
59
OPTICALSURFACEEXCITATIONS
We obtained a similar result for our NaF data, as shown in fig. 7. However, we have the problem that the damping function y(w) for NaF is not known at present and therefore has to be approximated, for instance by a Lorentz-function. In spite of the obvious improvement of the fit to the experimental points by using such a y(o)-function, there exist considerable theoretical problems concerning the evaluation of dispersion and linewidths
Wave vector
k
Fig. 7, Dispersion of surface polaritons in NaF. The theoretical curves I and II have been calculated using Sauter’s boundary conditions (see footnote of ref. 2), curve I with a constant damping y = 0.08 aTO, curve II with an approximated damping function I;(W).
to get ct (u) and ~~(0) in the case of frequency-dependent damping. For constant damping the linewidths of the ATR-spectra are a direct measure for the surface mode damping, which is essentially equal to the damping y of the bulk mode4v20). In fact, the experimental linewidths include further contributions arising from finite beam divergence and “radiative damping” by the prismls). We will omit here a detailed discussion of the difficulties associated with the evaluation of true linewidths, because of the theoretical problems mentioned in the footnote of ref. 2. From the results of Bryksin et al. and our own calculations, we conclude that the good agreement in the cases of GaP and CaF, is essentially due to the fact that the frequencies of the surface modes are in the region where the damping at room temperature is nearly constant. 3.3 SURFACE PLASMON-PHONON COUPLING We now want to turn to the effect of plasmon-phonon
coupling which
60
B. FISCHER,
N. MARSCHALL
AND
H. J. QUEISSER
can be studied very nicely in polar semiconductors. By changing the doping one can shift the plasma frequency wp through the longitudinal optical phonon frequency wLo. The frequencies of the coupled modes can be obtained from Raman scattering and reflectivity measurements 27, 2s). Theoretical considerations show that the optical surface waves should also be coupled. The coupled frequencies as function of carrier concentration have been calculated by Chiu and Quinn29). Wallis and Brion gave the dispersion relation for coupled surface modes30). Experimental work has been done by Genzel et al. on surface modes in small crystallites of Cd0 31), by Reshina, Gerbshtein and Mirlin on “semi-infinite” crystals of InSbsz), and by us on GaAsss). Reshina et al. obtained experimental points for nearly ten different coupling strengths. 4. Summary and conclusions We have considered experimental methods for optical excitation of nonradiative optical surface waves. The results presented are mainly concerned with the dispersion w(k) of these modes. In the long wavelength region dispersion and lifetime of the surface waves are determined essentially by the bulk parameters E,(U) and ~~(0). These parameters can be obtained from frequencies and linewidths of ATR-spectra; however, the evaluation of the linewidths is rather complicated, especially when the damping y is a function of frequency. A strongly frequency-dependent function y(w) influences also the dispersion relation as has been demonstrated for some alkali halides. Besides these bulk parameters, the influence of typical surface phenomena such as surface layers can be studied. Furthermore, the surface modes show also coupling between different types of excitations, plasmon-phonon and magnon-phonon coupling.
for instance
Acknowledgements We thank Professors R. J. Bell, H. Bilz, L. Genzel and J. J. Quinn for many interesting discussions. Financial support of the Fraunhofer Gesellschaft is gratefully acknowledged. References 1) R. Ruppin and R. Englman, Rept. Progr. Phys. 33 (1970) 149. 2) The derivation of the dispersion relation from Maxwell’s continuity conditions leads to eq. (l), where instead of EI(w) the complex dielectric function E(W)= EI(w) + isz (w) appears. [See for instance: M. Cardona, Am. J. Phys. 39 (1971) 1277.1 Difficulties with the formula including the complex E(W) arise from the fact that Maxwell’s continuity conditions are not complete for conducting media, which was shown by Sauter [F. Sauter, 2. Physik 203 (1967) 4881. Eq. (I) is a good approximation when
OPTICAL
3) 4) 5) 6) 7) 8) 9) 10)
I I) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33)
SURFACE
EXCITATIONS
61
~2< ~1. The derivation of the exact expressions for dispersion and lifetime of the surface waves, based on Sauter’s boundary conditions, will be published elsewhere. R. Fuchs and K. L. Kliewer, Phys. Rev. B 3 (1971) 2270. N. Marschall, B. Fischer and H. J. Queisser, Phys. Rev. Letters 27 (1971) 95. H. Boersch, J. Geiger and W. Stickel, Z. Physik 212 (1968) 130. J. B. Chase and K. L. Kliewer, Phys. Rev. B 2 (1970) 4389. H. Ibach, Phys. Rev. Letters 24 (1970) 1461. H. Raether, in: Springer Tracts in Modern Physics, Vol. 38, Eds. G. Hohler et al. (Springer, Berlin, 1965) p. 84. Y. Y. Teng and E. A. Stern, Phys. Rev. Letters 19 (1967) 511. R. H. Ritchie, E. T. Arakawa, J. J. Cowan and R. N. Hamm, Phys. Rev. Letters 21 (1968) 1530. D. Beaglehole, Phys. Rev. Letters 22 (1969) 708. J. Schoenwald, E. Burstein and R. F. Wallis, Bull. Am. Phys. Sot. [II] 16 (1971) 1409. A. Otto, Z. Physik 216 (1968) 398. C. M. Randell and R. D. Rawcliffe, Appl. Opt. 6 (1967) 1889. Physikalisches Institut der Universitlt N. Marschall, unpublished dissertation, Frankfurt/Main, Germany, 1971. W. E. Anderson, R. W. Alexander and R. J. Bell, Phys. Rev. Letters 27 (1971) 1057. H. Wolter, in: Hundbuch der Physik, Vol. 24, Ed. S. Fliigge (Springer, Berlin, 1956. p. 461. J. Hogglund and F. Sellberg, J. Opt. Sot. Am. 56 (1966) 1031. R. Ruppin, Solid State Commun. 8 (1970) 1129. N. Marschall and B. Fischer, Phys. Rev. Letters 28 (1972) 811. V. V. Bryksin, Yu. M. Gerbshtein and D. N. Mirlin, Fiz. Tverd. Tela 13 (1971) 2125 [Engl. Transl. Soviet Phys.-Solid State 13 (1972) 17791. V. V. Bryksin, Yu. M. Gerbshtein and D. N. Mirlin, Fiz. Tverd. Tela 14 (1972) 543. B. Fischer, N. Marschall and R. J. Bell, to be published. R. J. Bell, private communication, based on X-ray data. H. Bilz, L. Genzel and H. Happ, Z. Physik 160 (1960) 535. A. Mitsuishi, Y. Yamada and H. Yoshinaga, J. Opt. Sot. Am. 52 (1962) 14. A. Mooradian and A. L. McWhorter, in: Light Scattering Spectra of Solids, Ed. G. B. Wright (Springer, New York, 1969) p. 297. C. G. Olson and D. W. Lynch, Phys. Rev. 177 (1969) 1231. K. W. Chiu and J. J. Quinn, Phys. Letters 35A (1971) 469. R. F. Wallis and J. J. Brion, Solid State Commun. 9 (1971) 2099. L. Genzel and T. P. Martin, Surface Sci. 34 (1973) 33. I. I. Reshina, Yu. M. Gerbshtein and D. N. Mirlin, Fiz. Tverd. Tela 14 (1972) 1280. N. Marshall and B. Fischer, to be published.