Frequency Dependencies, Microwave Measurements

The very elementary relation between total and DC/AC conductivities can be expressed as follows  

That is to say, the total conductivity is a sum of tlie DC conductivity, which is a function of temperature, and the AC conductivity, which is function of frequency (f) and temperature. In this respect, the various conduction models predict varying relations for the AC conductivity. Fig. 6-14 shows the AC conductivity at 1 KHz and 1 GHz for p(3-octyl thiophene) (P(30T)), as a function for P(30T)-PE composites. For these materials, the AC conductivity is found to follow a simple relation, a = c where c is a constant and/is the frequency [123]. 

Although details of measurement methodology are provided in a subsequent chapter, we may note briefly here that for conductivities at frequencies less than 1 MHz (and fixed temperature), a capacitance-conductance bridge suffices. An impedance analyzer can be used to extend this range to 1 GHz. Microwave cavity perturbation techniques must be used in the range 10 GHz to 100 GHz. Kramers-Kronig transformation of Infrared Reflectance data can provide data for frequencies 100 GHz. 

Note the break in the abscissa for DC. The solid lines are drawn as a de for the eye. The data are for the sample used for Fig. 6-8. After Reference [188], reproduced with permission. 

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CONDUCTING POLYMERS Fundamentals and Applications 6.3.3 Frequency Dependencies, Microwave Measurements... 

Microwave measurements are typically performed at frequencies between 8 and 40 Gc/s. The sensitivity with which photogenerated charge carriers can be detected in materials by microwave conductivity measurements depends on the conductivity of the materials, but it can be very high. It has been estimated that 109-1010 electronic charge carriers per cubic centimeter can be detected. Infrared radiation can, of course, also be used to detect and measure free electronic charge carriers. The sensitivity for such measurements, however, is several orders of magnitude less and has been estimated to be around 1015 electronic charge carriers per cubic centimeter.1 Microwave techniques, therefore, promise much more sensitive access to electrochemical mechanisms. 

Although the conductivity change Aa [relation (8)] of microwave conductivity measurements and the Ac of electrochemical measurements [relation (1)] are typically not identical (owing to the theoretically accessible frequency dependence of the quantities involved), the analogy between relations (1) and (8) shows that similar parameters are addressed by (photo)electrochemical and photoinduced microwave conductivity measurements. This includes the dynamics of charge carriers and dipoles, photoeffects, flat band and capacitive behavior, and the effect of surface states. 

Intercollisional interference. We note that at the lowest frequencies the simple proportionality between absorption coefficient and product of gas densities breaks down. Under such conditions, certain many-body interactions affect the observations and modify the shape or intensities of the binary spectra, often quite strikingly. An example is shown in Fig. 3.3, a measurement of the absorption in a neon-xenon mixture in the microwave region, at the fixed frequency of 4.4 cm-1. Because of the frequency-dependent factor of g(v) that falls off to zero frequency as v2, absorption is extremely small at such frequencies, Eq. 3.2. As a consequence, it has generally been necessary to use sensitive resonator techniques for a measurement of the absorption at microwave frequencies... 

Abstract. Following a suggestion of Kostelecky et al. we have evaluated a test of CPT and Lorentz invariance from the microwave spectrosopy of muonium. Precise measurements have been reported for the transition frequencies U12 and 1/34 for ground state muonium in a magnetic field H of 1.7 T, both of which involve principally muon spin flip. These frequencies depend on both the hyperfine interaction and Zeeman effect. Hamiltonian terms beyond the standard model which violate CPT and Lorentz invariance would contribute shifts <5 12 and <5 34. The nonstandard theory indicates that P12 and 34 should oscillate with the earth s sidereal frequency and that 5v 2 and <5 34 would be anticorrelated. We find no time dependence in m2 — vza at the level of 20 Hz, which is used to set an upper limit on the size of CPT and Lorentz violating parameters. 

Using different microwave measurements of the conductivity from 3 up to 150 GHz (0.1 up to 5 cm-1), Donovan et al. [103] reported nearly the same temperature dependence at all frequencies between 300 and 20 K. Consequently, they found no evidence for a narrow dc collective mode implied by the FIR studies. The question of the existence of a strongly conductive mode is still open, but an accurate determination of the temperature dependence of the resistivity which may not have been achieved in all cases is a prerequisite before comparing dc and high-frequency data. 

Transient terahertz spectroscopy Time-resolved terahertz (THz) spectroscopy (TRTS) has been used to measure the transient photoconductivity of injected electrons in dye-sensitised titanium oxide with subpicosecond time resolution (Beard et al, 2002 Turner et al, 2002). Terahertz probes cover the far-infrared (10-600 cm or 0.3-20 THz) region of the spectrum and measure frequency-dependent photoconductivity. The sample is excited by an ultrafast optical pulse to initiate electron injection and subsequently probed with a THz pulse. In many THz detection schemes, the time-dependent electric field 6 f) of the THz probe pulse is measured by free-space electro-optic sampling (Beard et al, 2002). Both the amplitude and the phase of the electric field can be determined, from which the complex conductivity of the injected electrons can be obtained. Fitting the complex conductivity allows the determination of carrier concentration and mobility. The time evolution of these quantities can be determined by varying the delay time between the optical pump and THz probe pulses. The advantage of this technique is that it provides detailed information on the dynamics of the injected electrons in the semiconductor and complements the time-resolved fluorescence and transient absorption techniques, which often focus on the dynamics of the adsorbates. A similar technique, time-resolved microwave conductivity, has been used to study injection kinetics in dye-sensitised nanocrystalline thin films (Fessenden and Kamat, 1995). However, its time resolution is limited to longer than 1 ns. 

Figure 12.33 illustrates the set-up for LMMRS. The frequency response analyser replaces the single frequency lock-in amplifier used in the potential and light modulated microwave measurements described in Section 12.3. LMMRS detects the frequency-dependent modulation of the microwave reflectivity AR associated with the photogenerated minority carriers. This concentration decays by interfacial charge fransfer k d and recombination kKc)- The LMMRS response is therefore a semicircle with a characteristic frequency otam = + rec)- The low-frequency intercept of the... 

The lifetime of the trapped interchain polaron pairs responsible for the indirect non-radiative ODMR can be determined from the microwave chopping-frequency dependence of the in-phase and quadrature lock-in detected resonance signals [47]. Such measurements of the PL-enhancing polaron resonance of P3HT yielded a polaron pair lifetime t 5 /js, which is 10 times shorter than in PPVs [79]. The nature of this relatively short lifetime is not clear and obviously warrants additional investigation. Some speculations on this observation are offered in Section 2.1.3 below. 

Microwave measurements for PPy samples provide independent confirmation of the results reported at IR frequencies. Figure 15.46 compares (TdciT) with o-MwiT) for PPy(PF6) [29], PPy(TsO) [29], and PPy(S-PHE) [136]. The absolute values and the temperature dependence of cr in the dc and microwave frequency ranges for PPy(PF6) are nearly identical, in agreement with the Drude theory. The stronger temperature dependence of for PPy(TsO) and PPy(S-PHE) in comparison with cr wC T) is expected because the sample is in the localized regime [181]. 

Afc are different, the temperature dependence of these quantities is identical based on microwave measurements (Zhang et al. 1994). The frequency dependence of (Ti(ft)) is also very similar for the two polarizations. This indicates that die electrod5mamic response of YBa2Cu306 95 is identical along the a- and Z)-directions, except for the additional spectral weight along the a-axis. 

More can be learned when going to higher frequencies than those of microwaves. Figure 1.47 shows the frequency dependence of the conductivity of a polyacetylene sample doped with AsFs, measured in the entire frequency regime from d.c. up to terahertz (submillimeter waves, at room temperature) [103]. To cover such a wide frequency range, five different experimental methods had to be applied four-probe... 

The reaction of benzene cations with polymer chains was indicated by an increase in the electrical conductivity of the solution relative to that of neat 02-saturated benzene (see Figure 1.14). The increase in conductivity was more pronounced the longer the polymer chain length - an effect that was attributed to the fact that the charges diffuse over the entire length of the chains and encounter chain ends within one period of the oscillating electric field. Consequently, the measured intrachain conductivity was seen to be frequency-dependent with a tendency to increase with increasing microwave frequency [104]. 

The techniques used for measurement of AC conductivity are varied, and depend on the frequencies of interest. For v < ca. 10 Hz, a capacitance-conductance bridge can be used [396]. EIS, which yields both real and imaginary components of the complex resistance, i.e. impedance, can be used up to ca. 1 GHz. The cavity perturbation and related techniques in microwave measurements have been used for frequencies up to ca. 30 GHz [397]. For higher frequencies, mathematical (Kramers-Kronig) transformation of IR Reflectance data must be used [398]. 

Almost all microwave methods are non-contact [1] and allow simultaneous measurement of the magnitude and the frequency of vibrations. The distance between the inspected surface and microwave sensor can vary from several millimeters to a few meters. However, the accuracy of the measurement of vibration magnitude also depends on a distance between the microwave sensor and the object as well as the shape of the inspected surface. 

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