Surface plasmons and Sommerfeld–Zenneck waves on corrugated surfaces:

Surface Science 408 (1998) 203–211

Surface plasmons and Sommerfeld–Zenneck waves on corrugated surfaces: Application to high-T superconductors c D. Schumacher a,*, C. Rea a,b,1, D. Heitmann a, K. Scharnberg a a Institut fu¨r Angewandte Physik und Zentrum fu¨r Mikrostrukturforschung, Universita¨t Hamburg, Jungiusstraße 11, 20355 Hamburg, Germany b Department of Pure and Applied Physics, Queen’s University, Belfast, BT7 1NN, UK Received 18 September 1997; accepted for publication 26 February 1998

Abstract We have performed reflection measurements on corrugated YBa Cu O samples which exhibit below-gap surface plasmons and 2 3 7 Sommerfeld–Zenneck waves for temperatures below and above the critical temperature, respectively. We calculated the optical response of corrugated surfaces and in particular the coupling to surface plasmon polaritons and Sommerfeld–Zenneck waves. These calculations are in good agreement with our measurements. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Plasmons; Reflection spectroscopy; Superconductivity; Surface waves

1. Introduction Periodically corrugated surfaces in optical experiments act as grating couplers and allow coupling to dynamic surface excitations [1,2]. In particular, materials with negative real part e (v) r of the dielectric function e(v)=e (v)+ie (v) supr i port surface polaritons for e (v)%|e (v)|. For i r example, a metal at v=v (v plasma frequency) p p exhibits surface plasmon polaritons [3,4]. A material with optically active phonons supports surface phonon polaritons in the reststrahlen regime between the transversal and longitudinal phonon * Corresponding author. Fax: (+49) 40 41236332; e-mail: [email protected] 1 Present address: Seagate Technology (Ireland), 1 Disc Drive, Londonderry BT48 0BF, Northern Ireland. 0039-6028/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 98 ) 0 02 2 8 -3

frequency [5]. In this paper, we are particularly interested in the investigation of high-T materials. c A characteristic feature of a superconductor below the gap frequency v =2D/B is the large negative g real part of the dielectric function as compared with the imaginary part [6 ]. Under favorable conditions, these materials support ‘‘below-gap surface plasmons’’, i.e. plasmons with frequencies vT , is that the surface c plasmon resonances change into Sommerfeld– Zenneck waves [10].

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In the following we first give a brief introduction into grating couplers, we show experimental results on corrugated YBa Cu O surfaces, and then pre2 3 7 sent numerical calculations to analyze the observed optical response. We describe in detail the temperature dependence of the observed surface plasmon resonances in the regime of the critical temperature. Further, we will treat cap layers which allow us to enhance the signature of surface plasmon resonances in the experimental spectra. We discuss in particular the interplay of surface plasmon polaritons and Sommerfeld–Zenneck waves, impedance matching and diffraction effects.

2. Grating coupling to surface waves A negative real part e (v) and small imaginary r part e %|e | are the conditions for the existence of i r surface plasmon polaritons, whose behavior is intrinsically tied to the frequency response of the charge carriers in the supportive medium. Therefore any observation of this resonant electromagnetic mode will provide information on the dynamics of the carrier state in the conducting medium. Surface plasmons are the quanta of resonant electromagnetic fields which propagate along the surface (assumed in the x–y plane). They are associated with charge oscillation and evanescent fields. These fields decrease exponentially into the dielectric medium (z>0) that covers the surface and into the material with e <0 (z<0). Surface r plasmons on plane surfaces obey the dispersion relation (assuming vacuum as ambient dielectric) [1,2] v k = sp c

S

e(v) e(v)+1

.

enhance k . In the following we will consider the x former method. The gratings considered are translationally invariant in the y-direction ( Fig. 1) and the plane of incidence is the x–z plane. A grating of period a adds or subtracts integer multiples of the grating wave vector K to the momentum of the incident radiation 2p k =k +nK, K= , xn x a

(3)

where n=0, ±1, ±2, … are the diffraction orders. In particular, if k matches the plasmon disperxn sion, we expect resonant excitation of surface plasmons. The situation is sketched in Fig. 2 in the positions 1 and 2. We have performed such grating coupler experiments on polycrystalline YBa Cu O samples with 2 3 7 periodically corrugated surfaces. They are described in Section 3. Indeed, we found excitation of surface plasmons. However, the observed resonances are strongly governed by feedback of the grating coupler on the surface plasmon dispersion and by diffraction effects.

3. Experiments The experiments were performed on polycrystalline YBa Cu O samples. A corrugated surface 2 3 7

(1)

Since the real part of the in-plane surface plasmon wave vector k is always larger than v/c, surface sp plasmons do not couple directly with incident radiation which has in-plane wave vectors v k = sin h, x c

(2)

where h is the angle of incidence with respect to the surface normal. So grating coupler [1,2] or prism coupler techniques [3,4] are required to

Fig. 1. Sketch of the sample profile which is used for calculation. (a) The sample has a sinusoidally modulated surface with period a and height h. (b) Sample with additional Si cap layer.

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Fig. 2. Sketch of the momentum transfer by a grating coupler (frequency v vs. in-plane momentum k ). The dispersion relaxn tion of surface plasmons is shown by solid lines. The light lines (dotted lines) give the limit of direct momentum transfer from incident light. The dashed line shows the momentum of light, k , for an angle of incidence h=11°. Arrows 1 and 2 indicate x coupling to the surface plasmon resonance, 3 and 4 the onset of diffraction for larger frequencies.

was produced during the pressing of the YBa Cu O pellet before the tempering procedure, 2 3 7 analogous to the standard preparation procedure (see e.g. Ref. [11]). We used stainless steel dies with periodic rules of periods a=45–180 mm which had nearly triangular shapes of heights h= 5–45 mm. The critical temperature in our samples was T #90 K. The reflection measurements were c performed with a Fourier transform spectrometer (Bruker IFS-113v) in a flow-through cryostat with accurate temperature control. Wedged polyethylene windows allowed optical access. The sample holder had two positions to provide an in situ switching between the sample and a gold reference mirror. The angle of incidence was h=11°, the divergence of the beam ±3.5°. The detector was a 4.2 K germanium bolometer. A grid polarizer in the beam allowed polarization dependent measurements. Experimental spectra for a sample with h/a#15% are shown in Fig. 3. All spectra are normalized to the gold mirror. For both polarizations we measure a decrease of the reflectivity for wavenumbers n>100 cm−1. This is caused by the onset of the diffraction, which occurs for our near normal incidence experiment at v/c#2p/a, i.e. n#1/a [n is defined by n=1/l=v/(2pc)]. This

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Fig. 3. Reflectance measurement of a YBa Cu O sample with 2 3 7 corrugated surface. The p-polarized spectrum (solid line) has two absorption dips (A and B) which are not present in the s-polarized one (dashed line). A spectrum of a smooth sample is plotted as a dotted line. The smooth sample’s reflectivity reproduces data given in early papers on high-T superconducc tors [12]. The spectra for the corrugated sample contain additional absorption dips. Resonances labeled C to F are phonons (see e.g. ref. [13]).

situation is sketched in Fig. 2 in the positions 3 and 4. Since usually for a#l the p-polarized diffraction intensity is stronger than the s-polarized one (see e.g. ref. [14]), we have a stronger decrease for the p-polarized reflection. Several resonant features in Fig. 3 above n=120 cm−1 occur in both polarizations and are associated with phonons (resonances C to F ). The resonances marked A and B are observed only in p-polarization and can be identified as surface plasmon resonances. They will be discussed in the following section. Fig. 3 shows that the phonons leave complicated structures in the spectra already for the unstructured surface. The reason for these structures is the fact that the reststrahlen-type behavior, which is expected for an ideal Lorentz oscillator, is screened by the Drude part of the conductivity and also heavily influenced by the damping through e . This behavior is differently pronounced i in p- and s-polarized radiation, even at h=11°. The situation is even more complex with a grating coupler. The diffraction strength depends not only on the polarization, but also on e and e which r i vary strongly close to the phonon resonances, as

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known from the Lorentz oscillator model. This causes different situations and corresponding shapes of the phonon resonances in Fig. 3, which cannot easily be explained through simple models. We would like to make some comments on the use of polycrystalline samples. The requirements for surface plasmon polariton excitations are macroscopically homogeneous material on the scale of the plasmon wavelength which is about 100 mm in our experiment. This material also has to be homogeneously layered perpendicular to the interface. These conditions are well fulfilled for the pressed polycrystalline YBa Cu O samples which have 2 3 7 been investigated here.

4. Temperature dependence An interesting question for a superconductor is of course how the optical response differs above and below T . We measured a series of spectra at c different temperatures. Fig. 4 shows some typical reflection spectra, normalized to a spectrum measured at T=90 K. The sample had a grating period a=67 mm, and the temperature was varied between 10 K and 200 K. We find the well-known reflectiv-

Fig. 4. Temperature dependence of reflection enhancement below #220 cm−1 for a sample with grating period a=67 mm. The enhanced relative reflection [R(T )/R(90 K )] for T<90 K demonstrates the good superconducting quality of the structured sample. All spectra were taken for p-polarization. The arrows mark two phonons. However, surface plasmons are not visible due to resolution.

ity enhancement [12,15–17] R /R for supercond. normal v≤v #220 cm−1 (frequencies are given in units g of wavenumbers). For some time this v has g been identified with the superconducting gap v =3.5k T /B as expected from the BCS model. g B c This reflection enhancement measured on our corrugated surface demonstrates that they have the same superconducting quality as compared to the usual flat surfaces. Now we concentrate on a sample with grating period a=90 mm. We find, as shown in the inset to Fig. 5, that the surface plasmon resonance shows a significant dependence on T. For an accurate analysis of the resonance position, normalized spectra R /R were evaluated as shown on p s a widely expanded scale in the inset to Fig. 5. We have fitted the absorption dips with Gaussian profiles to determine the exact resonance position which is plotted in Fig. 5. We find three temperature regimes: (a) a slow increase with decreasing T from 300 K to T ; (b) a strong increase below c T which represents the transition from the normal c to the superconducting state; and (c) a flat (saturated ) regime for low temperatures. The resonance frequency increases sharply just below T . This is c

Fig. 5. Energy position of the first minimum of R /R (feature p s A in Fig. 3) plotted vs. temperature. The inset shows the ratio R /R for several temperatures (T=10 K to 110 K in steps of p s 20 K ), plotted vs. frequency (in wavenumbers) as solid lines. The fitted curves, which give the plasmon energy positions, are plotted as dashed lines. All curves are shifted in the y-direction for clarity. The minimum normalized reflectivity for all curves is 0.6±0.01.

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a clear sign of the fact that the transition to the superconducting state has a significant influence on the surface plasmon resonance.

5. Numerical calculations and application To explain our experimental results in more detail, we use a numerical algorithm based on the method of Chandezon et al. [18] and following work [19]. Chandezon et al. have developed a differential method for calculating the reflectivity and field distribution of multicoated gratings, which is capable of handling grating depths well in excess of the grating periods. Calculated spectra for a sample with a period a=90 mm are shown in Fig. 6. We use an expression given by Genzel et al. [12,20] to model the dielectric function e(v) of polycrystalline YBa Cu O . It consists of a sum of terms including 2 3 7 Drude type metallic behavior, superconducting response within a Mattis–Bardeen model [21], both connected in a two-fluid model, and some Lorentz oscillators describing the phonon contributions. With appropriate weights of these terms and parameters, it describes the reflectivity of the unstructured samples quite well (compare Fig. 3 with Fig. 6: the stronger intensity loss in the measurement of the unstructured sample at shorter

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wavelength is due to diffuse surface scattering). In our calculations we treat the surface as having a sinusoidal corrugation. We have also studied distorted sinusoidal gratings of the form w(x)=h(1 cos Kx+1)c, c>0.5 as well as profiles 2 2 where the grooves are separated by fairly wide flat stripes. The choice of the profile affects the dispersion relation [Eq. (1)] but the modifications found are rather small and would not be resolvable in our experiments because of the beam divergence. If we compare the calculated spectra in Fig. 6 with the experimental ones in Fig. 3, we find that the general behavior and most features are reproduced in the experiment. Differences in the details occur due to two effects. Firstly, the calculation assumes a perfect grating with homogeneous amplitude all over the sample, whereas the experimental sample has a certain variation in the amplitude. Secondly, the calculation assumes a parallel beam whereas in the experiments a divergent beam has to be used for intensity reasons. The averaging effect arising from the beam divergence is discussed in detail in Section 8. We would like to discuss in particular two effects. (i) In Fig. 6 we calculated that both R and R have nearly zero intensity at p s n #290 cm−1. Our calculations show that this m frequency of minimum intensity depends on the grating amplitude. Thus, if our sample is not perfectly homogeneous in amplitude, we do not get zero intensity, rather a broad flat minimum as observed in Fig. 3. (ii) From the calculations we find that the shape of the phonon related structures varies drastically with increasing grating amplitude. This behavior is partly due to grating coupler induced excitation of surface phonon polaritons. We do not elaborate on this behavior because we cannot resolve these sharp and detailed structures because of the averaging effect of the divergent beam.

6. Influence of the grating profile

Fig. 6. Calculated reflectivity spectra for a smooth sample (dotted line) and for a sample with corrugated surface for p(solid line) and s-polarization (dashed line).

To understand the experimental behavior in more detail, we analyze in the following several effects. We first investigate the coupling efficiency between the electromagnetic wave and the surface plasmon polariton. It depends primarily on the

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ratio h/a of grating height to grating period. Fig. 7 shows that there is a small and sharp dip for a weak modulation. This dip’s position, applying the grating relation Eq. (3), agrees very closely with the dispersion, Eq. (1), of the unstructured sample. With increasing modulation, this dip becomes deeper and at the same time broadens and shifts to smaller wavenumbers. For h/a=3%, the optimum condition is fulfilled, where the minimum is nearly zero. For stronger modulation the intensity minimum increases again, the resonance shifts and broadens further, losing its resonant character. This behavior can be explained in the following way: with increasing modulation, the plasmon couples more strongly with radiation, which causes both damping and a resulting broadening and feedback to the dispersion, resulting in a shift. If we use the language of quasi-particles for the coupled photon–plasmon quasi-particle, it can be said that the enhanced coupling to radiative photons decreases the real and increases the imaginary part of the quasi-particle self energy. The optimum condition for the deepest minimum can be considered as an impedance or power

Fig. 7. Influence of the modulation height h on the surface plasma resonance (using the experimental parameters for a and h). With increasing height h the minimum of the plasmon dip decreases and becomes close to zero at h/a#3%. For higher values of h/a, the dip broadens and the minimum value increases significantly. In addition, with increasing ratio h/a, the dip’s minimum position shifts to lower wavenumbers. Above #93.3 cm−1, the reflectivity drops due to the onset of the first diffraction order.

matching between the ‘‘internal’’ damping and the ‘‘external’’ radiative damping. The first arises from the imaginary part e and determines the linewidth i of the plasmon resonance at small h. The second process arises from the loss due to the grating coupler induced emission of radiation. Actually, for the optimal coupling, the width of the resonance is twice as large as compared to the ‘‘internal’’ width, which can be derived from Eq. (1), which neglects the effects of radiative losses. This behavior is very similar to that for prism arrangements and excitation of surface plasmons in silver films [1]. In that case, the coupling strength is determined by the width of the air gap in the Otto arrangement and by the thickness of the silver film in the Kretschmann configuration. Thus, for ideal experimental conditions one should work with a very shallow profile of about 3%. We will see, however, that in the real experiment due to the finite angular resolution these very sharp resonances smear out. We will discuss this below.

7. Sommerfeld–Zenneck waves We would also like to investigate another important aspect. From the formula used by Genzel et al. [12] to model the dielectric function e(v) of polycrystalline YBa Cu O , we find strong varia2 3 7 tions of the ratio e /|e | between the superconducti r ing and the normal state. The effect of this variation in e on the reflectivity is demonstrated in Fig. 8. For very small values of e , sharp resoi nances are found, which are, however, not very deep, since they do not meet the optimum condition discussed above. With increasing e and fixed i other parameters h and e , we first reach this r optimum condition. A further increase of e broadi ens the resonance and increases the minimum. Surprisingly, for very large values of e , we find i again a well pronounced minimum with a position very close to the light line. This situation, e &|e |, i r is actually the condition for Sommerfeld–Zenneck waves [10]. This surface electromagnetic wave, like the surface plasmon mode, only couples to p-polarized light. If we use the model function of Genzel et al. (we suppressed the so-called electronic oscillator

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Fig. 8. Influence of the imaginary part e . The plasma resonance i (SP, surface plasmon) regime of the p-polarized reflectivity is plotted for several values of e , constant e =−1500 and h/a= i r 3%. For large values of e , the resonance dip shifts towards the i diffraction onset. This is the Sommerfeld–Zenneck wave (SZW ) regime.

from Ref. [12]; it produces wrong values for e r in the normal state at small wavenumbers), we find for the superconducting state at the surface plasmon frequency e(v )=−1250+150i, thus sp |e |&e . In this case we have the conditions of a r i well-defined surface plasmon resonance. For the normal state we find e=−8+750i which is in the Sommerfeld–Zenneck regime. So in our experiments we actually switch between a more Sommerfeld–Zenneck type of excitation to a surface-plasmon excitation, if we vary the temperature from above to below T . c In detail we have the following behavior. For the optimum or smaller amplitudes a decrease in T and related decrease in e and increase of |e | i r causes a decrease of the resonance frequency, as shown in Fig. 8. For the large amplitude of h= 0.15a in our experimental situation, we have to consider both diffraction effects and the averaging effect of the divergent beam. In Fig. 9 we show that in the superconducting state with small e we i have a small plasmon dip, whereas in the normal state with large e we have a deep broad i Sommerfeld–Zenneck dip. Combined with the diffraction dip and the averaging effect of the

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Fig. 9. (a) Calculated reflectivity (p-polarization) for a corrugated surface, using a grating amplitude to grating period ratio of h/a=15%. In the superconducting state (solid line) the surface plasmon resonance is less pronounced than in the normal state (dashed line). (b) An average of the calculated reflectivity curves from (a) weighted with the angular resolution leads to broadened dips which approximately give the experimental width and shift of the resonance when T decreases below T . c

divergent beam (Fig. 9b), we calculate an upward shift in the superconducting state, as observed in the experiments.

8. Finite angular resolution So far we have not described the much broader linewidth in the experiment compared with calculations. For intensity reasons in our complex cryogenic setup we have to use a divergent beam. Consequently, the reflection measurement averages over a certain k regime. Since the plasmon resox nance closely follows the light line, we can directly translate this into an average on the frequency scale. Thus, our experiments not only see the plasmon resonance, but also the decrease of the intensity due to diffraction, in Figs. 6 and 7, above 93.3 cm−1. The sharp resonance of the surface

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plasmon smears out to a broad and less deep structure as demonstrated in Fig. 10. For future experiments we would like to make the proposal to shift the plasmon resonance away from the light line. This can be achieved by dielectric cap layers with high refractive index on top of the corrugated surface ( Fig. 1b). We expanded our algorithm to handle cap layers and show in Fig. 10 the influence of a Si layer which has e =11.8 and r no damping in the far infrared. Indeed, we find that a 5.0 mm (2.5 mm) layer of Si shifts the resonance by about 6 cm−1 (1 cm−1), which would be very helpful for experiments. The plasma resonance would remain resolvable from the diffraction onset, even if the averaging effect due to beam divergence is taken into account ( Fig. 10b).

9. Summary We have performed reflection experiments on corrugated high-T samples and show by comparic son with numerical reflectivity calculations that resonant excitations are governed by the interplay of surface plasmons, Sommerfeld–Zenneck waves and diffraction effects. In particular we find that at temperatures below T the high-T samples c c support below-gap surface plasmons and at T>T Sommerfeld–Zenneck waves. c Acknowledgements We acknowledge financial support from the Deutsche Forschungsgemeinschaft under Grant No. He 1938/4, the Deutscher Akademischer Austauschdienst, and the Graduiertenkolleg ‘‘Physik nanostrukturierter Festko¨rper’’.

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Fig. 10. Calculated broadening of the surface plasmon resonance. For an infinitely high angular resolution in (a) resonances are very sharp (full lines show resonance without cap layer). The finite angular resolution in (b) leads to broadening, in particular, it mixes the effect of the surface plasmon and the diffraction edge. The dotted (dashed ) lines are calculations with an assumed cap layer of 5.0 mm (2.5 mm) Si. This cap layer shifts the resonance away from the diffraction edge and allows a separation of both effects.

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