deletions in the structure of the maltose binding protein

Broadband microwave spectroscopy in Corbino geometry for temperatures down to 1.7 K Marc Scheffler and Martin Dressel Citation: Review of Scientific Instruments 76, 074702 (2005); doi: 10.1063/1.1947881 View online: http://dx.doi.org/10.1063/1.1947881 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/76/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Microwave resonances in dielectric samples probed in Corbino geometry: Simulation and experiment Rev. Sci. Instrum. 84, 114703 (2013); 10.1063/1.4827084 Broadband microwave spectroscopy in Corbino geometry at 3He temperatures Rev. Sci. Instrum. 83, 024704 (2012); 10.1063/1.3680576 Microwave induced zero-conductance state in a Corbino geometry two-dimensional electron gas with capacitive contacts Appl. Phys. Lett. 97, 082107 (2010); 10.1063/1.3483765 Broadband method for precise microwave spectroscopy of superconducting thin films near the critical temperature Rev. Sci. Instrum. 79, 074701 (2008); 10.1063/1.2954957 Strip-shaped samples in a microwave Corbino spectrometer Rev. Sci. Instrum. 78, 086106 (2007); 10.1063/1.2771088

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REVIEW OF SCIENTIFIC INSTRUMENTS 76, 074702 共2005兲

Broadband microwave spectroscopy in Corbino geometry for temperatures down to 1.7 K Marc Schefflera兲 and Martin Dressel 1. Physikalisches Institut, Universität Stuttgart, 70550 Stuttgart, Germany

共Received 22 February 2005; accepted 16 May 2005; published online 27 June 2005兲 We present a broadband microwave spectrometer covering the range from 45 MHz up to 20 GHz 共in some cases up to 40 GHz兲 which employs the Corbino geometry, meaning that the flat sample terminates the end of a coaxial transmission line. This setup is optimized for low-temperature performance 共temperature range 1.7–300 K兲 and for the study of highly conductive samples. The actual sensitivity in reflection coefficient can be as low as 0.001, leading to a resolution of 10% in absolute values of the impedance or complex conductivity. For optimum accuracy a full low-temperature calibration is necessary; therefore up to three calibration measurements 共open, short, and load兲 are performed at the same temperature as the sample measurement. This procedure requires excellent reproducibility of the cryogenic conditions. We compare further calibration schemes based on just a single low-temperature calibration measurement or employing a superconducting sample as a calibration standard for its normal state, and we document the capability of the instrument with test measurements on metallic thin films. Finally we apply the spectrometer to thin films of a heavy-fermion compound as an example for a strongly correlated electron system. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1947881兴

I. MOTIVATION

When studying the electronic properties of solids, optical spectroscopy in a general sense is a very versatile and powerful tool:1 the probe, the electromagnetic radiation, couples directly to the charge carriers, and the energy of the probe, corresponding to the frequency of the radiation, can be tuned to match the effects of interest. While the energy of typical band structures corresponds to visible and infrared light, the energy scale important for electronic correlations is typically much lower, like the far-infrared, submillimeter, or microwave regimes. Here one faces the low-frequency limit of geometrical optics: for frequencies below 30 GHz 共corresponding to ␭ = 1 cm兲, the wavelength exceeds typical sample sizes and is of the same order as typical dimensions of laboratory equipment. Thus for even lower frequencies, in the microwave range, completely different techniques have to be employed. Up to now, only a few broadband microwave experiments covering the particularly interesting range from 1 to 10 GHz have been performed on solids, and even fewer at lower temperatures. Thus we have developed a broadband microwave spectrometer that is particularly suited to study highly conductive solids in this frequency range and at low temperatures. The need for such an instrument in the study of correlated electrons, where temperatures below 10 K are required, is particularly obvious from recent microwave and optical studies on heavy-fermion compounds2–4 and the related Kondo insulators,5 but there are numerous other problems of solid state physics to be addressed in this frequency range like dynamical scaling6 and variable-range hopping7,8 in disordered systems, Luttinger liquids,9,10 Fermi a兲

Electronic mail: [email protected]

0034-6748/2005/76共7兲/074702/11/$22.50

liquids/non-Fermi liquids,11 quantum Hall effect,12,13 and Wigner crystallization14 in two-dimensional 共2D兲 electron systems, and magnetic resonance in the frequency domain.15 II. MICROWAVE SPECTROSCOPY

The only low-temperature technique to study solids at microwave frequencies that has been applied extensively in the past is the cavity perturbation technique.16,17 There the sample is introduced into a resonant cavity, and from the resulting change of resonance frequency and width, the complex sample properties can be determined. This technique is very sensitive with respect to small effects caused by the sample and is very robust with respect to the details of the microwave transmission lines required to fit a cavity into a cryogenic environment. However, usually only a single resonance is employed to study the sample, and thus no information concerning the frequency dependence is gained from measurements with just a single cavity. To actually obtain frequency-dependent data in the microwave range, several resonators 共cavities above 10 GHz as well as split-ring resonators18,19 for lower frequencies兲 can be combined, but since every frequency requires a different experimental setup, this is a rather arduous approach.20 One path to frequency-dependent measurements using just a single experimental setup is to excite a resonator in several different modes. The data will not be continuous in frequency, but at least several points can be obtained in some limited range. This is particularly easy if the resonator is based on a one-dimensional transmission line structure, like helical resonators21 and stripline or microstrip resonators.22–24 These resonators can usually be employed at around a dozen equidistant resonance frequencies. However,

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variable of the network analyzer is the complex reflection coefficient S11, the phase-sensitive ratio of the voltages of the reflected and outgoing waves. The reflection coefficient is determined by the impedance ZL of the load to be studied 共i.e., the sample兲 via the following relation:35 S11 =

ZL − Z0 , ZL + Z0

共1兲

where Z0 is the wave impedance of the coaxial cable, in our case the conventional standard of 50 ⍀. Since the sample dimensions are known, the sample impedance yields the complex material properties, for example the complex conductivity. In the case of a sample that is much thinner than the skin depth for all frequencies of interest, like for metallic thin films, the impedance of the Corbino disk is given by FIG. 1. 共a兲 Scheme of Corbino microwave spectrometer 共not to scale兲 and 共b兲 typical sample with inner and outer diameters a1 and a2 of Corbino disk.

up to now these techniques have only been applied in a few experimental studies, and in the case of the stripline and microstrip resonators these were basically limited to superconductors,22–24 whereas helical resonators were used to study dielectric materials.7 In contrast to the methods mentioned so far, we are interested in a truly broadband technique that continuously covers a large frequency range 共in particular including the range from 1 to 10 GHz that has hardly been studied兲 and can be employed at low temperatures. There are only two such techniques documented in the literature: first the Corbino approach that we also employ here, where the sample terminates a coaxial cable.25 This technique was pioneered by the group of Steven M. Anlage 共Maryland兲,26 and they used it to study thin films of high-temperature superconductors27,28 as well as ferromagnetic resonance and antiresonance features in CMR manganites.29,30 More recently the group of Mark Lee 共Virginia兲 has used a Corbino spectrometer31 to study Coulomb correlations in doped semiconductors.8 A rather different recent broadband approach uses the sample as a bolometer, i.e., the microwave absorption of the sample is determined by monitoring its temperature increase.32 Compared to the Corbino method, this technique is very sensitive and as such has been used to study single crystals of high-temperature superconductors.33 The disadvantage, however, is the lack of phase-sensitive information. III. CORBINO SPECTROMETER

For the design of the spectrometer, we have adopted general ideas of the Maryland group26 and developed numerous modifications to improve the accuracy and sensitivity by a modified calibration procedure and to reach much lower temperatures. The basic setup of the Corbino spectrometer is shown in Fig. 1共a兲. A microwave signal is created by the source of a network analyzer34 and fed into a coaxial transmission line. The signal is then reflected by the sample that terminates the otherwise open-ended coaxial cable at the Corbino probe. The reflected signal travels back through the same coaxial cable and is detected by the network analyzer. The output

ZL =

ln共a2/a1兲 , 2␲d ␴

共2兲

where d is the thickness of the sample, ␴ the complex conductivity, and a2 and a1 the outer and inner radii of the Corbino disk, respectively, as indicated in Fig. 1共b兲. In the case where bulk samples are studied that are much thicker than the skin depth, the load impedance is proportional to the surface impedance of the sample.26 To enable convenient room-temperature calibration of the setup using commercial calibration standards, the Corbino probe is manufactured from a commercial microwave adapter.36 For good mechanical and electrical contact of sample and probe, we use two springlike constructions: a massive spring presses the flat sample against the outer conductor of the probe. To bridge the gap between the sample plane and the end of the inner conductor, we use a small pin. The conical end of this pin in combination with resilient fingers of the inner conductor of the probe results in a small force that presses the pin against the sample. The general problem of such a Corbino setup is that the reflection coefficient measured by the network analyzer not only depends on the properties of the sample but also on undesired effects like the damping of the coaxial cable and partial reflections at microwave connections. To take these effects into account, one has to perform an extensive calibration, which will later be discussed in detail. This calibration sets a “reference plane;” in our case this is the plane of the Corbino probe. The load impedance ZL then comprises everything behind the reference plane, i.e., not only the impedance of the sample itself, but also any possible contact resistance between the Corbino probe and sample. Since there is no way to experimentally distinguish these two contributions, one has to minimize the contact resistance to effectively neglect it with respect to the sample impedance. In order to decrease the contact resistance and to precisely define the Corbino ring of the sample, we deposit metallic contact pads onto the sample 共typically gold, a few hundred nm thick兲 as indicated in Fig. 1共b兲. A. Cryogenics

As we are particularly interested in phenomena that occur at low temperatures, below 10 K, we have to incorporate the Corbino probe into a cryostat capable of 4He tempera-

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Broadband microwave spectroscopy

FIG. 2. Reproducibility of the microwave connections. The effect of disconnecting the microwave line between sample and calibration measurements is evident from this room-temperature measurement of the impedance of a bulk stainless steel sample. For clarity the “after” data are offset by 0.1 ⍀.

tures. Since we need a metallic coaxial cable to connect the low-temperature probe with the network analyzer located at room temperature, we face severe problems of heat transfer from the room-temperature environment into the cryostat. Here one has to consider a tradeoff: a longer cryostat with longer coaxial cables means lower base temperature, but it also means larger microwave losses finally leading to reduced sensitivity. The same tradeoff is valid for the material of the coaxial cables, whether copper or stainless steel. Finally we have chosen copper coaxial cables37 with a total length of 1.2 m from the network analyzer to probe. With a purpose-built glass cryostat 共four walls, helium and nitrogen volumes, plus intermediate isolation vacua兲 we reach a base temperature of 1.6 K by pumping on the helium bath. One crucial point that complicates the design of the cryogenic setup is the limited reproducibility of microwave connections: when disconnecting and reconnecting microwave coaxial connectors 共using the conventional procedures including torque wrenches兲, the transmission properties of the coaxial line cannot be reproduced completely. This effect is illustrated in Fig. 2: the impedance of a bulk stainless steel sample was measured at room temperature, once leaving all microwave connections untouched during the sample exchange necessary for calibration and in the second case disconnecting and reconnecting one microwave connection. While in both cases the overall impedance spectrum exhibits a square-root behavior that reflects the frequency dependence of the metallic surface impedance, the spectrum after reconnecting also shows a strong oscillatory contribution that is not present in the spectrum measured before reconnecting. This effect might be tolerable if lossy samples are studied 共where it appears less pronounced兲, but we are interested in low-loss samples and thus have to avoid it completely. Therefore we do not release any microwave connection within the course of sample and calibration measurements: we keep the complete microwave setup fixed with respect to the network analyzer. Instead, in order to exchange samples, we remove the cryostat from the probe insert by pneumatically lowering the entire glass cryostat as indicated in Fig. 3.

FIG. 3. Scheme of the overall setup of the spectrometer. To replace the sample, the glass cryostat is lowered and the stainless steel tube housing the exchange gas is removed. In order to reduce the overall length of highfrequency transmission lines, microwave source and test set are located as close to the top of the cryostat as possible. The test set converts the microwave signal to one of much lower, intermediate frequency 共if兲 which can then easily be transmitted to the 共more distant兲 actual network analyzer.

the sample. This is possible since the test set of the network analyzer offers an additional dc access to the coaxial cable, i.e., dc signals can be coupled into the coaxial cable independently of the microwave signal. This allows continuous twolead measurements of the sample resistance. This information is important for procedures described later, but additionally it serves to verify negligibility of the contact resistance 共if the real sample resistance is known from independent measurements兲.

C. Dual setup

Since the network analyzer set has two test ports and we are solely interested in reflection measurements, we can connect two independent Corbino probes to the network analyzer at the same time. This can be found in the general scheme of the apparatus in Fig. 3 as well as in the layout of the probe head shown in Fig. 4. Thus we can effectively measure two samples simultaneously during the same

B. Simultaneous dc measurement

An important tool to monitor the performance of the spectrometer is a simultaneous dc resistance measurement of

FIG. 4. Design of probe head incorporating two Corbino adapters.

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FIG. 5. General error model for a microwave reflection measurement.

cooldown. However, this means that two coaxial cables are incorporated into the insert, thus increasing the heat transfer and slightly increasing the base temperature.

source is reflected by the sample with actual reflection coefficient S11,a, whereas the reflection coefficient measured by the network analyzer is S11,m. Due to the three error contributions directivity ED 共signal reaches the detector directly without interacting with the sample兲, source match ES 共signal coming from the sample is reflected again and adds to the signal approaching the sample兲, and reflection tracking ER 共damping and phase shift in the transmission line兲, the actual and measured reflection coefficients differ, but there is a direct relation of the two as follows:

D. Temperature measurement and control

To enable convenient continuous measurements in the complete temperature range from 1.7 to 300 K, the probes are not immersed into the liquid helium bath, but instead are located inside of a stainless steel tube, which is surrounded by the helium liquid 共compare Fig. 3兲. For thermal contact, the tube is filled with helium gas at a pressure slightly lower than the vapor pressure at base temperature. The transmission properties of the coaxial cables strongly depend on temperature, e.g. the damping is mainly determined by ohmic losses of inner and outer conductors. Reproducibility of the spectrometer’s microwave characteristics therefore requires excellent reproducibility of the temperature distribution along the coaxial cables. Thus all parameters that contribute to the temperature distribution during a temperature-dependent measurement have to be reproduced. These are base temperature, helium and nitrogen levels, heater power, contact gas pressure, and relevant time spans. Thus we reproduce these settings right before the start of the measurements and then perform computer-controlled temperature-dependent measurements. For temperature measurement two Cernox sensors38 are mounted in the probe head as indicated in Fig. 4. Sensor A is located as close to the samples as conveniently possible and in combination with a resistive heater is used to control the temperature.39 During the acquisition of a microwave spectrum 共40 s兲, the temperature is stable within a few mK below 30 K. The presence of considerable temperature gradients is evident from sensor B located close to the heater. Therefore the temperature measured by sensor A cannot be regarded as the actual sample temperature either. To determine the actual sample temperature, we make use of the simultaneous dc resistance measurements: for several different samples with known temperature-dependent resistances, we have determined the actual sample temperature from the dc resistance. As the actual temperatures of all these independent measurements coincide for a given temperature measured by sensor A, we can establish a temperature correction for any measurement that strictly follows the procedures described above. IV. CALIBRATION A. General error model

Any broadband microwave experiment has to be calibrated. The general error model for a reflection measurement,40 combining any possible error sources, is shown in Fig. 5: the microwave signal coming from the

S11,m = ED +

ERS11,a . 1 − ESS11,a

共3兲

To obtain the actual reflection coefficient S11,a from the measured reflection coefficient S11,m, one has to know the three error coefficients 共which depend on frequency and temperature兲. These can be obtained from calibration measurements using three samples with known actual reflection coefficients. The calibration measurements are usually performed using a short 共ZL = 0兲, an open 共ZL = ⬁兲, and a load resistor, with real impedance 共conventionally ZL = 50 ⍀ is desired, but we choose smaller impedances as will be discussed later兲. B. Room-temperature calibration

Since the Corbino probes are modified commercial microwave adapters, the probes can be calibrated with commercial calibration standards at room temperature.41 This requires the center pin 关shown in Fig. 1共a兲兴, the sample holder, and its cylindrical housing 共shown in Fig. 4兲 to be removed from the probe. However, the probe head was designed in such a way that this removal leaves the coaxial cables 共and thus their transmission properties兲 untouched. The roomtemperature calibration can then serve for a large set of subsequent measurements 共we have used one for more than 100 low-temperature runs over a period of more than 1 year兲 without the need to disconnect the cylindrical housing or the center pin again. But even at room temperature additional calibration steps are necessary to account for the influence of the center pin that is required for actual sample measurements. C. Low-temperature calibration using three standards

For the calibration measurements at low temperatures, we use bulk aluminum or copper samples as short, teflon samples as open 共in contrast to leaving the probe empty, leading to a poorly defined position of the center pin兲, and NiCr thin films as load. Of these standards the appropriate load is most difficult to achieve. Here we modify the procedure proposed by Stutzman et al.31 NiCr is particularly suited for this task because it is metallic but has a comparably large dc resistivity, and furthermore the resistivity is almost temperature independent. This can be seen in Fig. 6 where the the dc resistance of several NiCr thin film samples is plotted as a function of temperature.42 In terms of the calibration procedure, the impedance of the load calibration sample is generally desired to match the 50 ⍀ wave impedance of the coaxial cable. However, we usually employ load samples with much lower resistance

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FIG. 6. Temperature dependence of the two-lead dc resistance for several NiCr thin film samples.

共typically between 10 and 20 ⍀兲 for the following reasons: First of all, the samples we want to study are mainly metallic and therefore the lower resistance of the load standard is closer to the actual samples. Second, with thicker films 共lower resistance兲 the influence of the substrate, in particular the intensity of dielectric resonances, is reduced. Third, with the dimensions of our Corbino probe, the thickness of a 50 ⍀ NiCr film is very close to the transition in film growth from islands to complete coverage and therefore hard to control.43 To avoid the influence of substrate resonances, we use glass substrates instead of sapphire and rather small sample sizes, effectively shifting the lowest substrate resonance to higher frequencies than usually accessed with the spectrometer. Resonances also have to be considered in the design of the open standard. To shift the lowest dielectric resonance frequency of the open standard to frequencies higher than those we are interested in, we have optimized the material and shape of the standard; finally we have settled for small cylindrical teflon samples. The calibration procedure requires known actual reflection coefficients of the calibration samples. While for open 共S11,a = 1兲 and short 共S11,a = −1兲 we assume perfect values,44 the actual reflection coefficient of the load sample is not known a priori since it depends on the details of the sample, in our case the resistivity and thickness of the NiCr film. Since the resistivity of the NiCr film is slightly temperature dependent 共typical changes between 2% and 4% between 1.6 and 300 K, compare Fig. 6兲, so is the actual reflection coefficient. Thus for each temperature where we perform a microwave measurement, we use the simultaneous dc resistance measurement to calculate the actual reflection coefficient, following Eq. 共1兲 where we set the load impedance ZL corresponding to the dc resistance of the NiCr film. As the dc resistance of the NiCr film is measured in a two-lead configuration, we have to subtract the contribution of the cables to obtain the actual resistance of the film. This cable contribution, in turn, is set as the dc resistance of the short calibration sample measured at the same temperature, i.e., we take the difference of the measured dc resistances of NiCr film and short standard as the actual dc resistance of the NiCr sample and assume that the impedance of the NiCr sample

Rev. Sci. Instrum. 76, 074702 共2005兲

FIG. 7. Temperature dependence of error coefficients at 10 GHz.

equals this resistance for the complete frequency range of the spectrometer. From this general low-temperature calibration procedure, one obtains the full frequency and temperature dependence of the error coefficients. In Fig. 7 the temperature dependence of the error coefficients is shown for an exemplary frequency. Here one can find that none of the error coefficients is much smaller than unity and therefore none can be neglected with respect to the reflection coefficient of the sample 关compare Eq. 共3兲兴. Furthermore, the error coefficients for this example change up to 0.4 as a function of temperature, i.e., the temperature dependence clearly cannot be neglected. One also sees that the details of cooling and heating strongly effect the error coefficients, most prominent in this example the sharp kink in ER at 5 K, when we have stopped to pump on the helium bath. The importance of carefully reproducing the temperature distribution is obvious when one keeps in mind that the final sensitivity in the reflection coefficient to be achieved with the spectrometer is 0.001, orders of magnitude smaller than the changes of the error coefficients with temperature. Stutzman et al. have proposed a slightly different lowtemperature calibration.31 They also perform three lowtemperature calibration measurements, but for the actual reflection coefficients they do not assume the ideal values 共e.g., S11,a = −1 for the short兲, but instead they use spectra that are determined experimentally at room temperature. The advantage of this strategy is that imperfections of the calibration samples could be incorporated. However, this requires that there is an independent, more precise room-temperature broadband spectrometer available for exactly the same frequency range as the cryogenic spectrometer to be calibrated, but that a room-temperature spectrometer also has to be calibrated with “known” samples, i.e., the problem of finding the appropriate samples is just shifted from one spectrometer to another. This technique only seems appropriate if for some reason the cryogenic spectrometer cannot be calibrated with “ideal” samples, but another, room-temperature spectrometer can. In our case we are able to apply almost perfect calibration samples to the cryogenic spectrometer directly, and therefore an additional step of using experimentally obtained room-temperature spectra for the actual low-temperature

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spectra does not enhance the overall accuracy of the calibration scheme.44 D. Low-temperature calibration using one standard

Using three reference samples is the reliable procedure to calibrate a microwave reflection measurement, and therefore measuring three calibration samples at low temperatures is an indisputable approach.45 However, the group of Steven M. Anlage has proposed to use a single low-temperature standard, the short, to determine only one temperaturedependent error coefficient ER and to use the roomtemperature values for ED and ES.26 The motivation was that the magnitude of ER 共at room temperature; a full lowtemperature calibration was not performed at that point兲 is larger than those of ED and ES and, more importantly, the latter two are mainly caused by the network analyzer and as such can be expected to be temperature independent. The data in Fig. 7 show that this approach roughly holds for our spectrometer, but in fact ED and ES are also temperature dependent, although weaker than ER. The room-temperature calibration data that is used in this case for ED and ES are not obtained using calibration “samples,” but with commercial calibration standards connected to the Corbino adapter. Thus these calibration measurements are performed without the center pin that connects the inner conductor of the probe with the sample. This change in probe geometry usually affects all three error coefficients, but in this “short only” procedure with only one additional measurement, its contribution 共as well as the temperature dependence of the spectrometer兲 is compensated only by ER. Therefore the resulting ER of the “three standards” and “short only” calibrations differ, although both are obtained from low-temperature calibrations. The “short only” calibration is only valid to measure samples with reflection coefficients close to that of the short calibration sample 共S11,a = −1兲 used for the low-temperature calibration. We verify this validity for our spectrometer as follows: for a range of 共real兲 impedances, the “measured” reflection coefficients S11,m was calculated using error coefficients obtained experimentally by “three standards” calibration. From these reflection coefficients the impedance was recalculated using the “short only” procedure 共using roomtemperature values of ED and ES and temperature-dependent ER obtained from “short only” calibration using the same short measurement as for the “three standards” calibration兲. The ratio of the impedances corresponding to the two calibration procedures is shown in Fig. 8 for a large range of possible sample impedances. Obviously the two procedures are consistent for small impedances, where the ratio becomes constant. Already here the ratio can differ from unity, indicating an incorrect experimental determination of the error coefficients. The sources of these errors are hard to pinpoint: they can stem from the short measurement that enters both calibrations, as well as from the open or load measurements that enter the “three standards” calibration, but also from the room temperature calibration that enters the “short only” calibration. The resulting error in impedance is of order 10% for high frequencies, giving a first estimate for the final error of the spectrometer including calibration.

FIG. 8. Consistency of “short only” 共so兲 and “three standards” 共3s兲 calibrations: ratio of the impedances obtained by the two procedures 共based on exemplary, experimentally determined error coefficients兲.

For larger impedances the impedance ratio for the two calibration schemes strongly deviates from unity. With a requirement that real and imaginary parts differ by less than 10% of the magnitude, the impedance to be studied using the “short only” procedure should not exceed 10 ⍀. This estimate is based on experimental data gathered with the current spectrometer and is governed by the errors that contribute during the calibration procedure. For a different, less reproducible spectrometer, this range might be reduced even further, and the validity of the “short only” calibration procedure has to be verified before it is applied. Thus the “short only” calibration is valid as long as highly conductive samples are studied, and in this case that procedure is more convenient because it only requires one additional low-temperature calibration measurement. For studies focusing on lossy or insulating samples, a modified procedure where ER at low temperatures is determined by measuring a single appropriate low-temperature standard like a load or an open might apply, but this was not investigated in the current project. E. Low-temperature calibration using a superconducting sample

In the case where one studies a sample at low temperatures that becomes superconducting at even lower temperatures, one can use the superconducting state of the sample as a short standard for the normal state of the same sample at slightly higher temperatures. Such a procedure assumes that the superconducting sample has an actual reflection coefficient S11,a = −1. This procedure works if the conductivity of the superconducting sample is much higher than that in the normal state for all frequencies of interest. While a superconductor by definition has zero resistance at zero frequency, for higher frequencies the conductivity decreases; at T = 0 it actually vanishes for frequencies below the superconducting energy gap.1 In practice the procedure to ensure that a superconducting sample can act as a short standard for the frequency range of interest was as follows: only if the results of a calibration with bulk aluminum or copper as a short and those of a calibration with the superconducting sample as a short match within the experimental errors, is the supercon-

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FIG. 9. Frequency dependence of two NiCr thin films 共thickness 10 and 25 nm兲, measured at 1.7 K.

ducting calibration considered to be valid. In this case the superconducting calibration can lead to an improved sensitivity for two reasons: first, since calibration and sample measurement are performed on the same sample during the same run, the contact resistance of the sample is present in both measurements and thus is included in the temperaturedependent error correction. Second, since both measurements are performed during the same cooldown, at almost the same temperature, and just a few minutes apart, the reproducibility of the microwave properties of the spectrometer can surpass that of the normal procedure where the spectrometer has to be warmed up, opened, and cooled down again. However, this only holds in the temperature range where the error coefficients are constant; for the example shown in Fig. 7 this is the case below 5 K. If the superconducting state of the sample can act as a short and additionally the “short only” scheme can be performed, the low-temperature calibration can be deduced completely from data of the same cooldown as the actual sample measurement. Obviously this can enhance the overall precision of the calibration procedure, and in addition it can increase the frequency range that can be reliably studied. For exceptionally suitable samples we could cover a frequency range of up to 40 GHz.

Rev. Sci. Instrum. 76, 074702 共2005兲

FIG. 10. Frequency dependence of the impedance of a 8 nm thick aluminium film, measured at 3 K. The calibration either employs bulk copper at 3 K 共Cu兲 or the superconducting sample at 1.7 K 共SC兲 as short. The two-lead dc resistance in the inset shows the superconducting transition.

data with those published by Stutzman et al.,46 we find improvements in frequency and temperature range as well as in accuracy: at 1.7 K we find a maximum change in the real part of the impedance of 10% up to 20 GHz 共6% up to 12 GHz兲, i.e., more than 30% improvement in the common frequency range, but much more important are the much larger frequency range 共70% increase兲 and the lower temperatures of these measurements 共1.7 instead of 4.2 K兲. To test the calibration procedure where the superconducting sample acts as a short, we have grown thin aluminum films. While bulk aluminum has a transition temperature of 1.19 K,47 which cannot be reached with our current setup, aluminum thin films exhibit enhanced transition temperatures.48 The superconducting transition at 2.0 K for an 8 nm thick aluminum film grown on sapphire can be found in the inset of Fig. 10. The impedance spectrum of this sample above Tc is shown in the main graph of Fig. 10, for two different calibrations: both use three standards, but in one case the short is bulk copper, whereas in the other case the short is the superconducting state of the sample at 1.7 K. In comparison, the superconducting calibration is slightly superior as evident in the imaginary part of the impedance which is closer to the expected value of zero.

V. TEST MEASUREMENTS

VI. APPLICATION TO HEAVY FERMIONS

To document the capability and sensitivity of the spectrometer, test measurements on known samples are required. In our case this is not so easy, since there hardly exist comparable spectrometers and thus no comparable measurements as well. Possible test samples are thin metallic films, where the impedance is expected to be constant 共and real兲 within our frequency range of interest and to match the dc impedance. Stutzman et al. have used NiCr thin films, similar to those employed as load standard for the calibration, to test their spectrometer.31 We have performed corresponding tests with our spectrometer, and the low-temperature impedance spectra of two NiCr films with different thicknesses are shown in Fig. 9. For both samples the real part of the impedance is almost constant, and the imaginary part is small compared to the real part, as expected. If we directly compare our

The first strongly correlated electron systems that we have studied with the Corbino spectrometer is UPd2Al3, one of the particularly well-studied heavy fermion compounds.49 In terms of the capability of the present Corbino spectrometer, UPd2Al3 is a perfect material to study: it is the only heavy-fermion compound where high-quality thin films were available and have been studied in detail.50,51 Here, in contrast to bulk samples, the impedance of a Corbino sample is increased considerably by the small sample thickness 共dramatically increasing the sensitivity of the spectrometer, as discussed later, to a value where actual measurements become reasonable兲 and we can work in the limit where the sample thickness is much thinner than the skin depth. Additionally, UPd2Al3 is superconducting below Tc = 2.0 K and we can employ the superconducting sample as a short calibration standard. Here we just present the complex conduc-

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FIG. 11. Conductivity spectra of UPd2Al3 at 3 K: comparison of “short only” calibration 共so, SC兲 using the superconducting sample as short and “three standards” calibration 共3s,Al兲 using bulk aluminum as short, teflon as open, and a NiCr thin film as load. Real and imaginary parts, ␴1 and ␴2, are vertically offset for clarity.

tivity spectra, whereas the detailed results concerning the conductivity of heavy fermions will be discussed elsewhere. Figure 11 shows the low-temperature conductivity spectrum 共real and imaginary parts兲 of UPd2Al3 for two different calibrations: in the first case the superconducting state was used as a short standard for the “short only” procedure, whereas in the second case the general “three standards” procedure was employed using aluminum, teflon, and NiCr standards. The main features can be found in the conductivity spectra obtained from both calibrations: ␴1 is almost constant for low frequencies and then exhibits a clear rolloff approaching zero at higher frequencies, whereas ␴2 vanishes for low frequencies and peaks at the frequency where the rolloff in ␴1 occurs, the so-called scattering rate. This general behavior is expected for a metal and follows the so-called Drude response.1 A characteristic for heavy fermions is the rather low frequency where this rolloff occurs, orders of magnitude lower than for normal metals. This strong reduction of the scattering rate was theoretically predicted for heavy fermions,1,52 but numerous optical studies on heavy fermions and a few additional cavity experiments searching for this feature could not observe it directly.53 The data presented in Fig. 11 clearly show the benefits of using the superconducting sample as a short compared to the conventional bulk aluminum sample. As the impedance of the sample at this temperature is approximately 0.15 ⍀, one is close to the low-impedance limit of the spectrometer to be discussed in the next section, leading to a decrease in sensitivity and the considerable scattering of the data. Thus the inherent advantages of the superconducting short are particularly appreciated, which are the inclusion of the contact resitance in the calibration measurement and the improvement in reproducibility of microwave transmission characteristics of the coaxial cable as evident from the considerable oscillations in the spectrum obtained with the aluminum short. VII. PERFORMANCE AND OUTLOOK

The measurements presented in the two previous sections clearly show the capability of our spectrometer. The standard temperature range is 1.7–300 K, the frequency range 45 MHz–20 GHz. In terms of the physical properties

Rev. Sci. Instrum. 76, 074702 共2005兲

FIG. 12. Contributions concerning the reproducibility of the spectrometer. Contact resistance: a bulk aluminum sample was measured at room temperature, removed completely, and measured again 共several times兲. The standard deviation 关sd共S11共RT兲兲兴 of these complex reflection coefficients is shown. Thermal cycling: The same short standard was measured from 1.7 to 300 K, cooled down again, and remeasured. We calculate the difference of the measured complex reflection coefficients at 1.7 K. This was repeated for five independent measurements of different samples. The average 关avg共兩S11,A共1.7 K兲 − S11,B共1.7 K兲兩兲兴 of these differences is presented. For comparison the standard deviation 关sd共S11共SC兲兲兴 of several measurements 共at 3 K兲 employing the superconducting calibration is shown.

to be studied, higher frequencies and lower temperatures are desirable. Concerning higher frequencies, above 20 GHz, we face problems with reproducibility that we were not able to solve with the present setup in a satisfactory way, except for the case of the superconducting samples mentioned above. These problems at higher frequencies are also present in room temperature measurements, and therefore we attribute them to the microwave components we use 共in particular the Corbino adapter and the other microwave connectors兲; there is little hope to overcome them in the near future. Concerning lower temperatures, improvements are possible within the current cryostat by employing stainless steel coaxial cables instead of copper ones 共during preliminary tests temperatures below 1.2 K were reached兲 and by incorporating the spectrometer into 3He or dilution refrigerators. The other important quantities that characterize the spectrometer are its sensitivity and its accuracy. The quantity we are interested in is the impedance of the sample. Thus we are interested in the sensitivity with respect to the impedance. This of course depends on the sensitivity of the reflection coefficient. The ultimate limit for detectable changes in the reflection coefficient is the network analyzer, but for our frequency range of interest this limit can be lower than 2 ⫻ 10−4 and can be neglected with respect to systematic errors, in particular concerning the reproducibility that is required for the calibration procedure. One source of systematic errors is the contact resistance between sample and Corbino probe. We estimate its effect as follows: at room temperature we measure a bulk aluminum sample, remove it completely from the probe, reassemble, and remeasure it. We repeat this procedure several times and calculate the standard deviation of the measured 共i.e., roomtemperature calibrated兲 reflection coefficients. The result is shown in Fig. 12: the error due to the contact resistance is around 0.001 below 1 GHz and increases for higher frequencies. Here one has to point out that this represents a worst-

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case scenario: the present sample represents a perfect short, and for any other sample the same contact resistance will have less impact on the reflection coefficient. The most stringent restriction for our experiment is the reproducibility of the low-temperature state of the spectrometer:54 if the temperature distribution along the coaxial cable differs for the calibration and sample measurements, the resulting changes of the transmission characteristics are not accounted for by the calibration. To quantify this error source, we consecutively perform two full temperaturedependent measurements from 1.7 to 300 K without changing the sample between the two cooldowns. We then calculate the difference of the 共room-temperature calibrated兲 reflection coefficients determined at 1.7 K, repeat this procedure for several samples, and determine the average difference, as presented in Fig. 12: below 1 GHz we observe a change of the reflection coefficient of 0.002, and at higher frequencies the error clearly increases continuously. This error sets the limit for standard low-temperature measurements. In Fig. 12 we also show that this limit is easily overcome by a low-temperature calibration using a superconducting sample: we have performed several independent lowtemperature measurements of the sample presented in Fig. 11 and determined the actual reflection coefficient at 3 K using the superconducting state at 1.7 K as a short standard for the “short only” calibration. The standard deviation of the resulting reflection spectra, presented in Fig. 12, is around or below 0.001 for frequencies up to 3 GHz and at least a factor of 2 smaller than the error discussed in the previous paragraph. This improvement of the superconducting calibration compared to the standard procedure is particularly evident at higher frequencies where the improvement can be larger than a factor of 5.55 Up to now we have only discussed the sensitivity concerning the measured reflection coefficient. Using Eq. 共1兲 one can now calculate the sensitivity in terms of the sample impedance ZL and find that the relative sensitivity ⌬ZL / ZL is a strong function of impedance, as is shown in Fig. 13. The sensitivity is optimal for ZL = 50 ⍀ and 共for our case assuming a sensitivity of 0.001 in the reflection coefficient兲 exceeds unity for ZL ⬍ 25 m⍀ and for ZL ⬎ 100 k⍀, i.e., there are clear bounds for meaningful measurements. When we take into account the error sources discussed up to now and additionally consider measurements on known samples, like those presented in the section on test measurements, and the comparison of different calibrations, like the consistency check for the “short only” procedure, we estimate a typical error in impedance 共and correspondingly in conductivity兲 of 10%. This value can easily compete with typical errors in conductivity data obtained by far-infrared and optical spectroscopy on metals and even with those of resonant cavities.1 Although the latter ones are reputed for their sensitivity, the accuracy in terms of sample conductivity is much poorer due to problems to account for sample shape and field distribution. Apart from the lower temperatures already discussed, other modifications of the spectrometer are possible. A magnet with the potential to study magnetic resonance phenom-

FIG. 13. Relative sensitivity with respect to the load impedance assuming a constant sensitivity of 0.001 with respect to the reflection coefficient.

ena in frequency space30 can be easily incorporated into the cryostat. In this range of frequency, ferromagnetic and antiferromagnetic resonances are of particular interest. Furthermore, although our interest was focused on metallic thin films, the spectrometer can be directly applied to other samples, in particular less conductive bulk samples like those of heavily doped semiconductors8 or Kondo insulators.5 In conclusion, we have developed a broadband microwave spectrometer operating in the frequency range from 45 MHz up to 20 GHz 共in special cases even up to 40 GHz兲 and a temperature range from room temperature down to 1.7 K. Due to improved design and calibration methods and particularly reproducible operation procedures we reach considerably enhanced accuracy as well as range of operation leading to a very wide scope of possible applications.

ACKNOWLEDGMENTS

The authors thank Gabriele Untereiner for the preparation of the calibration and test samples and Martin Jourdan and Hermann Adrian 共Universität Mainz兲 for the UPd2Al3 samples. We appreciate valuable discussion with Andrew Schwartz and Steven M. Anlage, and we acknowledge helpful communications with Marcy Stutzman and Mark Lee. The project was funded in part by the Deutsche Forschungsgemeinschaft 共DFG兲. APPENDIX A: THREE ADDITIONAL CALIBRATION MEASUREMENTS

In the most general case, three calibration measurements of known samples are performed and from these three measurements the three error coefficients are obtained, i.e., three reflection coefficients Sm,a , Sm,b, and Sm,c are measured from samples with known actual reflection coefficients Sa,a , Sa,b, and Sa,c, respectively. Then three equations Sm,j = ED +

ERSa,j 1 − ESSa,j

共A1兲

can be solved for ED , ES, and ER, and these three error coefficients can be determined as a function of the Sm,j and Sa,j. Using a common denominator N defined as

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共A2兲

N = Sm,aSa,aSa,c − Sa,aSm,aSa,b − Sa,aSm,cSa,c − Sm,bSa,bSa,c + Sa,bSm,cSa,c + Sa,aSa,bSm,b , the results for the error coefficients are:

共A3兲

ES · N = Sa,bSm,c − Sm,cSa,a + Sm,aSa,c − Sm,bSa,c − Sm,aSa,b + Sa,aSm,b , ED · N = − 共− Sa,aSm,aSm,bSa,c + Sa,aSm,aSa,bSm,c + Sa,aSm,cSm,bSa,c + Sm,aSa,cSa,bSm,b − Sm,aSa,cSa,bSm,c − Sa,aSm,cSa,bSm,b兲,

共A4兲 2 2 2 2 2 2 2 2 2 2 2 2 Sm,a Sm,bSa,b − Sm,cSm,b Sa,b Sa,a − Sm,cSm,a Sa,a Sa,b − Sa,a Sm,aSm,b Sa,b − Sm,a Sa,b Sm,bSa,a + Sm,aSa,b Sm,b Sa,a ER · N2 = Sa,a 2 2 2 2 2 2 2 2 2 2 2 2 − Sa,a Sm,c Sa,bSm,b + Sa,a Sm,cSa,bSm,b − Sm,cSa,c Sm,bSa,b − Sm,c Sa,c Sm,aSa,b − Sm,c Sa,c Sm,bSa,a + Sm,c Sa,c Sm,bSa,b 2 2 2 2 2 2 2 2 2 2 2 2 − Sm,cSa,c Sm,aSa,a + Sm,c Sa,c Sm,aSa,a − Sa,a Sm,a Sm,bSa,c − Sm,aSa,cSa,b Sm,b − Sm,bSa,b Sa,cSm,c − Sm,aSa,a Sa,cSm,c 2 2 2 2 2 2 2 2 2 2 2 2 + Sm,cSm,b Sa,b Sm,c + Sm,cSm,a Sa,a Sa,c − Sm,a Sa,cSa,b Sm,c + Sa,aSm,a Sm,bSa,c − Sa,aSm,aSm,b Sa,c + Sa,a Sm,aSm,b Sa,c 2 2 2 2 2 2 2 2 2 2 2 2 − Sa,aSm,aSa,b Sm,c + Sa,a Sm,aSa,bSm,c + Sa,aSm,a Sa,b Sm,c + Sa,a Sm,c Sm,bSa,c + Sa,aSm,cSm,b Sa,c − Sa,a Sm,cSm,b Sa,c 2 2 2 2 2 2 2 2 2 2 2 2 − Sm,a Sa,c Sa,bSm,b + Sm,aSa,c Sa,bSm,b + Sm,a Sa,cSa,b Sm,b + Sm,aSa,cSa,b Sm,c + Sm,a Sa,c Sa,bSm,c + Sa,aSm,c Sa,b Sm,b .

APPENDIX B: ONE ADDITIONAL CALIBRATION MEASUREMENT

ES −

If only one low-temperature calibration measurement is performed, one has to assume that the general temperature dependence of the three error coefficients can be satisfactorily replaced by letting just one error coefficient ER be temperature dependent. This procedure can convert the data obtained by the network analyzer 共NWA兲 using the room temperature calibration directly, without the need to recalculate the raw data measured by the NWA. The NWA uses the error coefficients ED , ES, and ER,NWA 共obtained by conventional room-temperature calibration兲 to determine the NWA-calibrated reflection coefficient SNWA from the measured raw reflection coefficient Sm SNWA =

Sm − ED . ER,NWA + ES共Sm − ED兲

共B1兲

But at low temperature the actual reflection tracking ER,a differs from the room temperature reflection tracking ER,NWA and thus the measured Sm 共raw data兲 is determined as follows 共with Sa being the actual reflection coefficient of the sample兲: Sm = ED +

ER,aSa . 1 − E SS a

共B2兲

Combining Eqs. 共9兲 and 共10兲 one obtains SNWA =

ER,aSa . ER,NWA + SaES共ER,a − ER,NWA兲

共B3兲

To determine the low-temperature reflection tracking ER,a, one performs an additional low-temperature measurement with a known standard with actual reflection coefficient stand , whereas the NWA obtains SNWA 共including roomSstand a temperature calibration兲, i.e., Eq. 共B3兲 now reads stand = SNWA

ER,aSstand a ER,NWA + Sstand ES共ER,a − ER,NWA兲 a

.

共B4兲

From such a measurement the low-temperature ER,a can be obtained:

ER,a = ER,NWA ES −

共A5兲

1 Sstand a 1

.

共B5兲

stand SNWA

In the current project a short was used as the lowtemperature standard with actual reflection coefficient = −1. Thus Eq. 共B5兲 becomes Sshort a ER,a = ER,NWA

ES + 1 . 1 ES − short SNWA

共B6兲

If the actual reflection tracking is obtained like this then the actual reflection coefficient S11,a of an arbitrary sample can be obtained from Eq. 共B3兲: Sa =

SNWAER,NWA . ER,a + SNWAES共ER,NWA − ER,a兲

共B7兲

1

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Krzystek, J Telser, A. Mller, C. Sangregorio, D. Gatteschi, and M. Dressel, Phys. Chem. Chem. Phys. 5, 3837 共2003兲. 16 O. Klein, S. Donovan, M. Dressel, and G. Grüner, Int. J. Infrared Millim. Waves 14, 2423 共1993兲; S. Donovan, O. Klein, M. Dressel, K. Holczer, and G. Grüner, ibid. 14, 2459 共1993兲; M. Dressel, O. Klein, S. Donovan, and G. Grüner, ibid. 14, 2489 共1993兲; M. Dressel, O. Klein, S. Donovan, and G. Grüner, Ferroelectrics 176, 285 共1996兲; H. W. Helberg and M. Dressel, J. Phys. I 6, 1683 共1996兲. 17 M. Mola, S. Hill, P. Goy, and M. Gross, Rev. Sci. Instrum. 71, 186 共2000兲. 18 W. N. Hardy and L. A. Whitehead, Rev. Sci. Instrum. 52, 213 共1981兲. 19 D. A. Bonn, D. C. Morgan, and W. N. Hardy, Rev. Sci. Instrum. 62, 1819 共1991兲. 20 A. Hosseini, R. Harris, Saeid Kamal, P. Dosanjh, J. Preston, Ruixing Liang, W. N. Hardy, and D. A. Bonn, Phys. Rev. B 60, 1349 共1999兲. 21 R. J. Deri, Rev. Sci. Instrum. 57, 82 共1986兲. 22 M. S. DiIorio, A. C. Anderson, and B.-Y. Tsaur, Phys. Rev. B 38, 7019 共1988兲. 23 D. E. Oates, A. C. Anderson, and P. M. Mankiewich, J. Supercond. 3, 251 共1990兲. 24 N. Belk, D. E. Oates, D. A. Feld, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 53, 3459 共1996兲. 25 O. M. Corbino, Nuovo Cimento 1, 397 共1911兲. 26 J. C. Booth, D. H. Wu, and S. M. Anlage, Rev. Sci. Instrum. 65, 2082 共1994兲. 27 D. H. Wu, J. C. Booth, and S. M. Anlage, Phys. Rev. Lett. 75, 525 共1995兲. 28 J. C. Booth, D. H. Wu, S. B. Qadri, E. F. Skelton, M. S. Osofsky, A. Piqué, and S. M. Anlage, Phys. Rev. Lett. 77, 4438 共1996兲. 29 A. Schwartz, M. Scheffler, and S. M. Anlage, Phys. Rev. B 61, R870 共2000兲. 30 A. Schwartz, M. Scheffler, and S. M. Anlage, cond-mat/0010172 共2000兲. 31 M. L. Stutzman, M. Lee, and R. F. Bradley, Rev. Sci. Instrum. 71, 4596 共2000兲. 32 P. J. Turner et al., Rev. Sci. Instrum. 75, 124 共2004兲. 33 P. J. Turner et al., Phys. Rev. Lett. 90, 237005 共2003兲. 34 Vector network analyzer set HP 85107B by Hewlett-Packard. 35 D. M. Pozar, Microwave Engineering 共Wiley, New York, 1998兲. 36 Type 08 S 121-K00 S3, by Rosenberger Hochfrequentechnik. The female connector was converted to a Corbino probe by removing most of the threading on a lathe. Although the remaining part of the threading can still be used to connect room-temperature calibration standards or the cylindrical housing shown in Fig. 4, it does not interfere when the sample is attached to the mating plane of the connector. 37 Type UT85-LL-TP, by Micro-Coax. 38 Model CX-1050-AA, by LakeShore. 39 Model 340 temperature controller, by LakeShore. 40 Taschenbuch der Hochfrequenztechnik, edited by K. Lange and K.-H. Löcherer 共Springer, Berlin, 1992兲.

41

2.4 mm Calibration Kit HP 85056A, by Hewlett-Packard. Since the resistance was obtained from the two-lead measurement that is performed simultaneously with respect to the microwave measurements, the measured values contain a resistance contribution due to the cables 共approximately 1.5 ⍀兲 which is slightly temperature dependent and causes part of the temperature dependence of the resistance plotted in Fig. 6. 43 Thermal evaporation of commercial NiCr pellets, product number 042468 by Alfa Aesar. 44 The only standard where we could determine deviations from perfect behavior is the teflon open. From spectra calibrated commercially at room temperature, we can interpret the open standard as a capacitance of ⬇30 fF. But the measurements presented here focus on low-resistive samples where the reflection coefficient is very close to the short standard 共S11 = −1兲, and thus the influence of the open standard 共S11 = 1兲 is comparably small. We have found no improvement when incorporating the imperfection of the open standard and therefore neglect it. For highly resistive samples on the contrary one clearly has to take this effect into account. 45 H. C. F. Martens, J. A. Reedijk, and H. B. Brom, Rev. Sci. Instrum. 71, 473 共2000兲. 46 Our 28 ⍀ film corresponds to the 40 ⍀ film of Stutzman et al. as their Corbino probe has different diameters. 47 W. Buckel, Supraleitung, 5th ed. 共VCH, Weinheim, 1994兲. 48 W. Buckel and R. Hilsch, Z. Phys. 138, 109 共1954兲; R. W. Cohen and B. Abeles, Phys. Rev. 168, 444 共1968兲; M. Strongin and O. F. Kammerer, J. Appl. Phys. 39, 2509 共1968兲; H. Sixl, J. Gromer, and H. C. Wolf, Z. Naturforsch. A 29A 319 共1974兲. 49 C. Geibel et al., Z. Phys. B: Condens. Matter 84, 1 共1991兲. 50 M. Huth, A. Kaldowski, J. Hessert, Th. Steinborn, and H. Adrian, Solid State Commun. 87, 1133 共1993兲. 51 M. Jourdan, M. Huth, and H. Adrian, Nature 共London兲 398, 47 共1999兲. 52 A. J. Millis and P. A. Lee, Phys. Rev. B 35, 3394 共1987兲. 53 The only previous experiments probing the conductivity of heavy fermions around the scattering rate were on UPt3 and employed cavities: P. Tran, S. Donovan, and G. Grüner, Phys. Rev. B 65, 205102 共2002兲. For a review on optical studies on heavy fermions see: L. Degiorgi, Rev. Mod. Phys. 71, 687 共1999兲. Meanwhile several other heavy-fermion compounds have been studied using conventional far-infrared spectroscopy, but none could directly reveal the Drude rolloff. 54 Obviously this only applies to low-temperature measurements. For measurements at room temperature the contact resistance discussed before is the dominant error source. 55 Between the measurements the sample was removed completely and different contact pads were deposited. Thus part of the differences documented in terms of the standard deviation might stem from differences in the sample and not from the spectrometer. The data shown here therefore act as a worst-case bound for the performance of the calibration procedure employing the superconducting sample. 42

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