Surface plasmons and Sommerfeld–Zenneck waves on corrugated surfaces:: Application to high-Tc superconductors
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 ϵr(ω) of the dielectric function ϵ(ω)=ϵr(ω)+iϵi(ω) support surface polaritons for ϵi(ω)≪|ϵr(ω)|. For example, a metal at ω≲ωp (ωp plasma frequency) 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 frequency [5]. In this paper, we are particularly interested in the investigation of high-Tc materials. A characteristic feature of a superconductor below the gap frequency ωg=2Δ/ℏ is the large negative 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 ω<ωg. These below-gap surface plasmons have recently been reported on high-Tc material by Dunmore et al. [7]and on Al films in the microwave regime by Buisson et al. [8]. A first evidence for below-gap surface plasmons was also reported in ref. [9]. Another feature we found in our experiments, as ϵi(ω) becomes larger than |ϵr(ω)| in the relevant frequency regime for T>Tc, is that the surface plasmon resonances change into Sommerfeld–Zenneck waves [10].
In the following we first give a brief introduction into grating couplers, we show experimental results on corrugated YBa2Cu3O7 surfaces, and then present 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.
Section snippets
Grating coupling to surface waves
A negative real part ϵr(ω) and small imaginary part ϵi≪|ϵr| are the conditions for the existence of 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
Experiments
The experiments were performed on polycrystalline YBa2Cu3O7 samples. A corrugated surface was produced during the pressing of the YBa2Cu3O7 pellet before the tempering procedure, analogous to the standard preparation procedure (see e.g. Ref. [11]). We used stainless steel dies with periodic rules of periods a=45–180 μm which had nearly triangular shapes of heights h=5–45 μm. The critical temperature in our samples was Tc≈90 K. The reflection measurements were performed with a Fourier transform
Temperature dependence
An interesting question for a superconductor is of course how the optical response differs above and below Tc. We measured a series of spectra at 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 μm, and the temperature was varied between 10 K and 200 K. We find the well-known reflectivity enhancement 12, 15, 16, 17Rsupercond./Rnormal for ω≤ωg≈220 cm−1 (frequencies are given in units of
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 μm are shown in Fig. 6. We use an expression given by Genzel et al. 12, 20to model
Influence of the grating profile
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 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.
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 ϵ(ω) of polycrystalline YBa2Cu3O7, we find strong variations of the ratio ϵi/|ϵr| between the superconducting and the normal state. The effect of this variation in ϵ on the reflectivity is demonstrated in Fig. 8. For very small values of ϵi, sharp resonances are found, which are, however, not very deep, since they do not meet the optimum condition discussed
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 kx regime. Since the plasmon resonance 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
Summary
We have performed reflection experiments on corrugated high-Tc samples and show by comparison 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 Tc the high-Tc samples support below-gap surface plasmons and at T>Tc Sommerfeld–Zenneck waves.
Acknowledgements
We acknowledge financial support from the Deutsche Forschungsgemeinschaft under Grant No. He 1938/4, the Deutscher Akademischer Austauschdienst, and the Graduiertenkolleg “Physik nanostrukturierter Festkörper”.
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