Dressel.Gruner.Electrodynamics of Solids.2003

R

Fig. 11.12. Schematic of lumped resonance circuit. The source is weakly coupled to the resonator by the inductance L. Rx and Cx represent the sample. The tunable capacitance Cp allows coverage of a wide range of frequency.

11.3.1 Resonant circuits of discrete elements

In the audio and radio frequency range, up to approximately 100 MHz [Hil69], RLC resonant circuits are used to enhance the sensitivity of dielectric measurements. The complex dielectric constant is evaluated from the change in the resonance frequency and the decrease of the quality factor upon introducing the sample. The sample is usually contained between two parallel plates and is modeled by a parallel circuit of a capacitive part Cx and a resistive part Rx. Fig. 11.12 shows a simple corresponding circuit; as the capacitance Cp is varied, the signal detected by the voltmeter goes through a maximum when the resonance condition is passed (cf. Section 9.3). Roughly speaking, the resonance frequency depends on the dielectric constant 1 of the material introduced into the capacitor, while the width of the resonance curve increases as the losses of the sample (described by 2) increase. Following this arrangement, resonant circuits are designed to operate over a large range of frequencies; they are capable of high accuracy, provided the losses are low.

11.3.2 Microstrip and stripline resonators

As we have seen in Section 9.3, any transmission line between two impedance mismatches forms a resonant structure. Microstrip and stripline resonators utilize the fact that the resonance frequency and bandwidth of the transmission line resonator depend upon the electrodynamic properties of the conductors and of the dielectric media comprising the transmission line. Consequently, measurement configurations using microstrip resonators offer the following: if a conductor has to be studied, the stripline itself is made out of the material of interest; insulating material, on the other hand, is placed as dielectric spacers between the metallic

Microstrip resonators are a well established technique used to study superconducting films which are deposited onto a substrate to form a resonant microstrip pattern; and in Fig. 11.13 measurements on a YBa2Cu3O7−δ superconducting stripline resonator operating in the vicinity of 3 GHz [Lan91] are displayed. The phase velocity vph, a parameter proportional to the resonance frequency ω0, is plotted as a function of the temperature; this parameter also contains the penetration of the electromagnetic field into the structure – and thus is a measure of the penetration depth λ. The phase velocity vph(T ) increases for decreasing temperature because λ(T ) becomes larger [Zho94].

11.3.3 Enclosed cavities

In the frequency range from 1 to 300 GHz enclosed cavities are employed to measure the dielectric properties of materials; the sample is introduced into the cavity and the changes of the resonance parameters are observed [Don93, Dre93, Kle93, Sch95]. Enclosed cavities are not limited by diffraction problems which other methods face if the wavelength λ becomes comparable to the sample size – a common occurrence in this range of frequency. Often it is sufficient to consider the sample (placed in the cavity) as a small perturbation of the resonator; the material parameters are evaluated from this perturbation as outlined in Section 9.3.3. The major disadvantage of cavities – like any resonant technique – is that they are (with a few exceptions) limited to a single frequency.

If the material under study is a low-loss dielectric, the entire cross-section of a cavity can be filled by the sample, as shown in Fig. 11.14a. The evaluation of the complex dielectric constant ˆ is straightforward [Har58, Hip54a] since the geometrical factor is particularly simple. If the sample is a good conductor (and significantly thicker than the skin depth), it can replace the wall of a cavity. In Fig. 11.14b a cylindrical TE011 cavity is displayed where the copper end plate is replaced by a sample. The advantage of these two configurations is that the analysis has no geometrical uncertainties. Instead of filling the entire cavity or replacing part of it, a small specimen (with dimensions significantly smaller than the dimensions of the cavity) can be placed inside the cavity. This also leads to a modification of the cavity characteristics, which can be treated in a perturbative way. For a quantitative evaluation of the electrodynamic properties of the material, however, the geometry of the sample is crucial. Details of the analysis and discussion of particular sample shapes are found in [Kle93, Osb45, Pel98].

The experimental results obtained from a specimen which undergoes a metal– insulator transition at 135 K are shown in Fig. 11.15. The needle shaped crystal of α-(BEDT-TTF)2I3 was placed in the electric field antinode of a 12 GHz cylindrical TE011 reflection cavity. In Fig. 11.15a we display the temperature dependence of

The reflected and transmitted intensities of a symmetrical Fabry–Perot resonator are described by the Airy function plotted in Fig. B.3. Instead of the quality factor, the finesse F defined as

is commonly used to describe the resonator; here is the half-intensity full width of the transmission maxima of the Airy function 1/[1 + F sin2{β}] which peak at β = 2π d/λ = ±mπ , with m = 0, 1, 2, . . . accounting for higher harmonics. The finesse is related to the quality factor by Q = 2νdF (with the frequency ν = ω/(2π c) = 1/λ) and can be as large as 1000.

The simplest open resonator is built of two plane parallel mirrors; to reduce radiation losses the mirrors have to be much larger than the wavelength used. Spherical and hemispherical Fabry–Perot resonators were also developed [Cla82, Cul83] which overcome limitations due to these diffraction effects and due to alignment problems. In a good approximation the quality factor is given by

Q = ω0d/(1 − Reff)c, where d is the distance between the mirrors and Reff is the effective reflectivity of the open resonator. At 150 GHz, for example, quality

factors up to 3× 105 have been achieved, making this a very sensitive arrangement. We distinguish between three different arrangements which are used to measure material properties by open resonators. First, a slab of a low-loss material is placed inside the resonating structure. Second, the dielectric sample itself forms a Fabry–Perot resonator where the light is reflected at the front and at the back of the specimen. Third, one of the mirrors is replaced by the (highly conducting) sample. In the first arrangement, the electrodynamic properties of a low-loss material are determined by introducing it into the resonator and measuring the change in frequency and halfwidth of the resonance; some of the possible experimental setups are shown in Fig. 11.16. For these arrangements the evaluation of the complex conductivity by the perturbation method is less accurate compared to that

in enclosed cavities because in general radiation losses cannot be neglected.

In the second case, the sample itself forms the resonant structure – for example, a slab with the opposite sides being plane parallel; due to the impedance mismatch at the boundaries, multireflection within the sample occurs. The experiments, which are conducted either in transmission or reflection, map the interference pattern in a finite frequency range; fitting this pattern yields electromagnetic properties of the dielectric material using the expressions given in Appendix B. This method requires that the sample dimensions significantly exceed the wavelength – the thickness being a multiple of λ/2. The upper frequency limit is given by the ability to prepare plane surfaces which are parallel within a fraction of a wavelength. Due to these limitations, the technique of using the sample as a Fabry–Perot

(b)

Sample

In

Out

Fig. 11.16. Open resonator setups for the measurement of the optical conductivity: (a) confocal transmission resonator with low-loss specimen in the center; (b) hemispherical type for highly conducting samples.

resonator is mainly employed in the submillimeter and infrared frequency range. The measurements are performed by placing the Fabry–Perot arrangement in an optical spectrometer; the data are taken in the frequency domain or by Fourier transform technique. In combination with a Mach–Zehnder interferometer, the real and imaginary parts of the response can be measured independently, as shown in Fig. 10.1.

In order to measure highly conducting samples – the third case – the specimen is used as part of the resonant structure (e.g. as one mirror); most important is the case of thin films deposited on low-loss dielectric substrates. The (usually transparent) substrate acts as an asymmetric Fabry–Perot interferometer with one mirror made of the thin film. The interference pattern depends on the real and imaginary parts of the electrodynamic response of the film and of the substrate. If the latter parameters are known (for example by an independent measurement of the bare substrate), the complex conductivity σˆ of the conducting film is evaluated from the position and the height of the absorption minima. The detailed analysis of the optical properties of a sandwich structure is given in Appendix B. When the optical properties of a bulk sample – instead of a thin film – are investigated, a transparent material is brought in contact with the specimen; the measurements are performed in reflection configuration. Interference minima appear when the optical thickness of the dielectric is roughly equal to an integer number of a half

[Ram93] S. Ramo, J.R. Whinnery, and T.v. Duzer, Fields and Waves in Communication Electronics, 3rd edition (John Wiley & Sons, New York, 1993)

[Ros90] A. Roseler,¨ Infrared and Spectroscopic Ellipsometry (Akademie-Verlag, Berlin, 1990)

[Sch95] A. Schwartz, M. Dressel, A. Blank, T. Csiba, G. Gruner,¨ A.A. Volkov, B.P.

Gorshunov, and G.V. Kozlov, Rev. Sci. Instrum. 60 (1995)

[Sri85] S. Sridhar, D. Reagor, and G. Gruner,¨ Rev. Sci. Instrum., 56, 1946 (1985)

[Tom93] H.G. Tompkins, A User’s Guide to Ellipsometry (Academic Press, Boston, MA, 1993)

[Zho94] S.A. Zhou, Physica C 233, 379 (1994)

Further reading

[Alt63] H.M. Altschuler, Dielectric Constant, in: Handbook of Microwave Measurements, edited by M. Sucher and J. Fox, 3rd edition (Polytechnic Press, New York, 1963), p. 495

[Azz91] R.M.A. Azzam, ed., Selected Papers on Ellipsometry, SPIE Milestone Series MS 27 (SPIE, Bellingham, 1991)

[Ben66] H.E. Bennett and J.M. Bennett, Validity of the Drude Theory for Silver, Gold and Aluminum in the Infrared, in: Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abeles` (North-Holland, Amsterdam, 1966)

[Cha85] S.H. Chao, IEEE Trans. Microwave Theo. Techn. MTT-33, 519 (1985)

[Col92] R.E. Collin, Foundations For Microwave Engineering (McGraw-Hill, New York, 1992)

[Coo73] R.J. Cook, in: High Frequency Dielectric Measurements, edited by J. Chamberlain and G.W. Chantry (IPC Science and Technology Press, Guildford, 1973), p. 12

[Duz81] T. van Duzer and T.W. Turner, Principles of Superconductive Devices and Circuits (Elsevier, New York, 1981)

[Gri96] J. Grigas, Microwave Dielectric Spectroscopy of Ferroelectrics and Related Materials (Gordon and Breach, Amsterdam, 1996)

[Her86] G. Hernandez, Fabry–Perot Interferometers (Cambridge University Press, Cambridge, 1986)

[Hip54b] A.v. Hippel, Dielectric Materials and Applications (Technology Press M.I.T., Wiley, Cambridge, 1954)

[Jam69] J.F. James and R.S. Sternberg, The Design of Optical Spectrometers (Chapman and Hall Ltd, London, 1969)