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ISSN 0020-4412, Instruments and Experimental Techniques, 2006, Vol. 49, No. 5, pp. 669675. © Pleiades Publishing, Inc., 2006. Original Russian Text © A.F. Shevchun, M.R. Trunin, 2006, published in Pribory i Tekhnika Eksperimenta, 2006, No. 5, pp. 8289.

ELECTRONICS AND RADIO ENGINEERING

A Method for Measuring the Surface Impedance of Superconductors in the Temperature Range 0.4120 K
A. F. Shevchun and M. R. Trunin
Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: shevchun@issp.ac.ru
Received March 14, 2006

Abstract--A method and a setup for making precision measurements of the temperature dependences of components of surface impedance Z(T) = R(T) + iX(T) of small superconductor crystals in the temperature range 0.4120 K have been developed. The setup combines a high-quality-factor resonance system for measuring the microwave response of a sample at a frequency of 28 GHz and a refrigerating unit with the use of evaporation of 3He vapors. Measurements of Z(T) of optimally doped YBa2Cu3O6.93 single crystals in the superconducting and normal states were performed to illustrate the operation of the setup. PACS numbers: 74.25.Nf, 07.57.Pt, 07.20.Mc DOI: 10.1134/S0020441206050101

INTRODUCTION Taking measurements of the temperature dependences of surface impedance Z(T) = R(T) + iX(T) in absolute units at microwave frequencies = 2f is one of the experimental methods for studying superconducting materials. The real part of the impedance--surface resistance R(T)--is related to energy losses of an electromagnetic wave reflected from a superconductor. The imaginary part--reactance X(T)--determines the nondissipating energy stored in the surface layer of the sample under study. The requirements for the accuracy of microwave measurements became more stringent with the development of the physics of superconductors. Only precise microwave techniques allowed measurements of the dependence Z(T) in classical superconductors with Tc 10 K [1, 2] to be taken. These measurements were very informative: the value of energy gap was found from temperature dependences of surface resistance R(T) ~ e/T and reactance X(T) ~ e/T at T < Tc/2; field penetration depth (T) into a superconductor, from X(T) = µ0(T) at T < Tc; and the mean free path of electrons, from measurements of R(T) and X(T) in the normal state at T Tc. In the case of local electrodynamics, the complex conduction of a superconductor can easily be found from impedance components measured in absolute units: (T) = '(T) i''(T) = iµ0/Z2(T). The possibility of applying the BardeenCooperSchrieffer (BCS) theory to explanation of the properties of classical superconductors was confirmed by a nonmonotonic behavior (a coherent peak) of the real part of microwave conductivity '(T) observed at temperatures of 0.8 < T/Tc 1 [3, 4]. The issue of the accuracy of microwave measurements became more pressing after the advent of high-

temperature superconductors (HTSCs), in which highquality single crystals are of small dimensions. One of the features of HTSCs is that their temperature dependence Z(T) does not follow the BCS theory: instead of an exponential function at low temperatures, linear temperature dependences of both the surface resistance and the field penetration depth are observed [5], which are typical of superconductors with the d symmetry of the order parameter. The method proposed in [6] is the most convenient for measuring the surface impedance of samples having surfaces no larger than ~1 mm2. The main idea of this method is that a sample is placed on the end of a dielectric rod at the center of a cylindrical resonator operating on the H011 mode, i.e., into the region of an almost uniform magnetic microwave field. The resonator is evacuated in order to provide the possibility of varying the rod's temperature without changing the temperature of the resonator's walls. A sapphire rod is commonly used, because this material is characterized by a high thermal conductivity and weakly absorbs microwaves. By changing the rod's temperature and measuring the resonance frequency and Q factor of the resonator in the presence and absence of a sample, it is possible to determine the temperature dependences of both components of the sample's surface impedance. The complexity of the experimental problem can be inferred from the example considered at the end of this paper. To measure the temperature dependence Z(T) of an YBa2Cu3O6.93 (YBCO) sample at a frequency f = 3 в 1010 Hz at T < 30 K, it is necessary to reliably record change f = 100 Hz in the resonance frequency at a change in the sample temperature of 1 K; i.e., f / f ~ 109 (upon superconducting-to-normal-state transition of

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the sample, the resonance frequency changed by 300 kHz). The solution to this problem required the use of a high-Q resonator. First, we used a superconducting niobium resonator the walls of which were cooled with liquid 4He from the outside. Its Q factor without a sample but with a sapphire rod placed inside it was Q0 = 1.8 в 106 at a temperature of the resonator's walls of T = 4.2 K. At T = 1.4 K, the Q factor was Q0 = 3.2 в 106. Second, a highly stable Agilent PSG-L E8244A generator with a frequency resolution of up to 0.1 Hz and an absolute stability no worse than 1 kHz/day was used. To preclude the effect of pickups from the supply line, the generator was powered through a voltage regulator. Measurements began several hours after the generator had been turned on. Third, a constant temperature of the resonator's walls to within an accuracy of 0.1 K was ensured during temperature changes of the rod with the sample. Evacuation of the resonator allowed Z(T) measurements to be taken at temperatures of 0.4120 K. A sample was cooled to temperatures below 2 K through evacuation of vapors of liquid 3He by an external cryogenic pump [7]. A copper heat sink on which the sapphire rod was mounted was immersed directly into liquid 3He. DESCRIPTION OF THE SETUP Figure 1 shows the part of the insert for the cryostat that is immersed into liquid 4He. The insert is assembled from thin-walled (0.30.5 mm) stainless-steel tubes joined with argon welding to adapters and flanges from the same steel. The insert can conventionally be subdivided into two parts: a resonator unit and a sample-cooling unit. Cylindrical niobium cavity resonator 1 constitutes the main part of the resonator unit. It is a cylinder with two caps. Its inner diameter and height are 14 mm. The H011 mode at a frequency f = 28.2 GHz was excited in the resonator. Since this is a degenerate mode with respect to the E111 mode, in order to suppress and shift E111 relative to H011 along the frequency axis, there are cylindrical protrusions with a diameter and height of 2.5 mm. Two holes with a diameter of 1.6 mm in the upper cap spaced by 7 mm are used to input microwave signals into and output from the resonator. Two rectangular waveguides 3 with an internal cross section of 3.4 в 7.2 mm2 that transmit microwave signals are placed in tube 2 and terminate in corner junctions (E-corner-type bents of the waveguide line). The short, 8-mm-long, sides of these corners are soldered to a brass washer supporting them. Holes drilled in the brass washer and corners coincide with the holes in the resonator's upper cap. Thin indium gaskets (not shown in Fig. 1) laid around these holes preclude the direct inleakage of a microwave signal between the brass washer and the resonator's upper cap. Coupling elements 4--coaxial waveguide with loops at their ends--

are inserted into the channels formed. The size of the coupling loops was selected experimentally: the total length of the central conductor is 7.5 mm, 3.5 mm of which belongs to the loop. Teflon parts of the coaxial waveguides are fixed in links 5 transforming the rotational motion of regulators, which are placed outside the cryostat at room temperature, into the translational motion of the coupling elements. The degree of coupling between the microwave channel and the resonator can be varied smoothly during an experiment by changing the distance between the loops and resonator. The bottom cap of the resonator is pressed to its cylindrical part by bellows 6 serving as a spring. A 1mm-diameter sapphire rod (7) enters the resonator through a 2-mm-diameter hole positioned at the center of the bottom cap. Sample 8 is fixed with Apiezon vacuum grease at the rod's upper end. RuO2thermometer 2 and heater 10 are attached to the part of the sapphire rod located outside the resonator. Six 0.05-mm-diameter manganin wires laid along thick-walled tube 11, which ensures the rigidity of the structure, are connected to the thermometer and heater. The sapphire rod is glued into a copper holder, which is pressed to the end of copper rod 12 with a brass nut. A copper rod 10 mm in diameter and 110 mm in length is soldered to a collector of liquid 3He (13). Tube 2, the inner parts of the waveguides and resonator, and the upper part of the copper rod constitute a common vacuum cavity, which is evacuated before the experiment through tube 11. After the insert is cooled, the residual gas is frozen out and a high vacuum establishes in the cavity. All dismountable joints in the chain of components that includes flanges I and II, the cylindrical part of the resonator, and the flanges of parts III and IV are sealed hermetically with indium gaskets and clamped with brass bolts. To ensure access to the sapphire rod, the brass bolts joining the flanges of parts III and IV are unscrewed and the resonator unit suspended by tube 2 is lifted upward, while the sapphire rod, the resonator's bottom cap, and part IV remain at their places. Being supplied to the sample-cooling unit, gaseous 3He passes through 40- and 25-mm-diameter tubes 14 and 15, respectively. When 4He is evacuated from the cryostat's cavity, which is cooled to T = 1.4 K, gaseous 3He condenses on the walls of these tubes. The condensate from tube 15 enters the collector of liquid 3He and cools the copper rod. A vacuum jacket consisting of tubes 16 (30 mm in diameter and 80 mm in length), 17 (14 and 60 mm), 18 (18 and 50 mm), and 19 (80 and 80 mm) and upper and bottom caps is fastened to tube 15 with a silver solder. During its manufacture, the vacuum jacket was carefully evacuated to a high vacuum. To maintain vacuum at a low temperature, an adsorbent (activated carbon) is placed on the bottom cap. The gaseous 3He used in the sample-cooling unit is stored at room temperature in a 25-l vessel at a 0.6-atm pressure (Fig. 2). The amount of gas contained in the
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17 18 IV 12 13

Stainless steel Niobium Brass Copper Indium gaskets 19 Vacuum

Fig. 1. Design of the insert's part placed inside the cryostat: (1) niobium resonator, (2) tube, (3) waveguides, (4) coupling elements, (5) links, (6) bellows, (7) sapphire rod, (8) sample, (9) thermometer, (10) heater, (11) thick-walled tube, (12) copper rod, (13) collector of liquid 3He, (14, 15) 3He-supplying tubes, (1619) tubes of the vacuum jacket, and (IIV) parts of the resonator unit. INSTRUMENTS AND EXPERIMENTAL TECHNIQUES Vol. 49 No. 5 2006

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Computer NI PCI-6014 ADC board GRIB

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Amplifier Microwave detector Gasholder To the helium line

Agilent E8244A Microwave oscillator

to 4He-evacuating pump

Manometer B2 B1 Thermometer LakeShore 340 Temperature controller Cryostat Heater Liquid 4 He Gaseous 3 He

Cryosorption pump

Fig. 2. Measurement diagram. V1 and V2 are valves.

vessel is monitored with a manometer. The vessel is joined to the cryogenic pump via a 10-mm-diameter bellows hose through valve V1. The pump is manufactured from a thin-walled stainless-steel tube 22 mm in diameter and 90 cm in length welded from below and filled with activated carbon as the sorbent. Copper radiators installed inside this tube improve the heat exchange between the sorbent and helium bath. An КМи-40 portable Dewar flask is used to cool the pump. The volume of the cryogenic pump is large enough to evacuate the entire 3He contained in the insert. Valve V2 mounted at the top of the pump is connected to the insert via a long flexible bellows hose 40 mm in diameter. Because the resonator placed inside the cryostat is immersed in liquid 4He, a change in the vapor pressure above the liquid leads to a change in the pressure on the resonator's walls, which in turn causes a change in the dimensions of the resonator. It has been ascertained experimentally that a change in the 4He vapor pressure in the cryostat by 1 Torr leads to a change in the resonance frequency by 200 Hz. Therefore, to maintain a constant pressure P 1.05 atm inside the cryostat, an oil gasholder having a volume of 50 l is used. As the gas fills the gasholder, the surplus amount of gas is discharged into the helium line. Three operating temperature ranges of the setup can be distinguished: (i) at temperatures of 5 < T < 120 K, the vapor pressure above 4He in the cryostat is maintained at a constant level with the gasholder;

(ii) at temperatures of 2 < T < 5 K, gaseous 4He is evacuated from the cryostat's cavity with a mechanical pump to a pressure of 2 Torr, which corresponds to a temperature of liquid 4He of 1.4 K; and (iii) at temperatures of 0.4 < T < 2 K, gaseous 4He condenses in the collector and is evacuated by the cryopump. Evacuating 4He vapors allows the temperature in the cryostat's cavity to be reduced from 4.2 to 1.4 K within 45 min; during this period, the entire amount of 3He condenses in the collector. Evacuating 3He vapors with cryosorption pump allows the temperature of the liquid to be lowered from 1.4 to 0.65 K within 2 min and from 0.65 to 0.5 K within 20 min. The operating time in this mode is 4 h. A further decrease in the temperature requires that the residual gas be removed from the cryopump and repeated evacuation be performed. As a result the temperature of 3He falls from 0.5 to 0.4 K within 40 min. According to the data from [8], the thermal conductivity of the sapphire rod 1 mm in diameter and 10 mm in length contained in the resonator is ~1.5 µW/K at T = 0.4 K. This estimate imposes stringent limitations on the microwave power absorbed by a sample: a microwave power no higher than 100 nW must be dissipated in the resonator at T < 1 K.
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THE MEASUREMENT CIRCUIT AND DATA PROCESSING Figure 2 shows a block diagram of the automated system for measuring the temperature dependences of the Q factor and the resonance frequency. The temperature of the sapphire rod is controlled with a LakeShore 340 controller that measures the resistance of the RuO2 thermometer, which is installed on the sapphire rod, according to a four-probe circuit at a frequency of 20 Hz. Each measurement of the Agilent 82357A is recorded by a computer (PC) through a GPIB adapter, while the parameters of the proportionalintegralderivative temperature-regulating algorithm are inputted into the controller. A 300- wire resistor attached to the sapphire rod serves as a heater. Through the GPIB adapter, the PC specifies the frequency and power of the signal radiated by the Agilent PSG-L E8244A generator; this signal propagates along the waveguide and is fed to the resonator. A microwave transmitted through the resonator passes through a waveguide and falls on a measuring waveguide detector with a D-607 diode, which operates in the quadratic mode. The detector output signal passes through a laboratory amplifier with a gain of 5000 and arrives at the board of an NI PCI-6014 ADC, which inputs the measured voltage into the PC at a 200-kHz frequency and 16-bit resolution. After the required temperature of the rod is set, the frequency dependence of the microwave power transmitted through the resonant system is recorded, as sweeping frequency fsw of the generator-radiated signal is varied and the voltage across the diode is measured. Measurements of the impedance of superconducting samples begin at a low temperature and are performed at minimum levels of coupling between the microwave line and the resonator, at which the maximum accuracy of low-loss measurements is attained. In this case, the frequency dependence of the microwave power transmitted through the resonance system has the form of an ordinary resonance curve: Q L P0 P ( f sw ) = ----------------------------------------------------- , 2 2 2 4 Q L ( f 0 f sw ) / f 0 + 1
2

to a value that ensures a signal level sufficient for measurements at T > Tc. If a substantial part of a coupling loop is in the resonator, the resonance curve observed cannot be described by formula (1). This is determined by the fact that the detector-registered microwave signal passes not only through the resonator but also through an additional channel--directly from one coupling loop to another. Assuming that, near the resonance peak in the resonance curve, the phase of a direct-penetration signal remains constant, the dependence of the microwave power arriving at the diode on frequency fsw can be written in the form A0 P ( f sw ) = ------------------------------------------------- + Be 2 Q L ( f 0 f sw ) / f 0 + i
i 2

,

(2)

where A0 is the amplitude of the resonance signal and B and are the amplitude and phase of the direct-penetration signal, respectively. The curves obtained under different coupling levels are joined via subtraction of constants 1/Q and f , which are temperature independent, as was purposefully checked, from the corresponding values of 1/QL(T) and f (T) measured at large coupling values. The temperature dependences of surface resistance R(T) and reactance X(T) for an isotropic sample can be found with the perturbation theory from the relationships [9]: R ( T ) = [ Q ( T ) Q0 ( T ) ] ;
1 1

2 X ( T ) = ------ [ f ( T ) f 0 ( T ) ] + X 0 , f

(3)

(1)

where P0 is a constant independent of frequency fsw, f0 is the resonance frequency, and QL is the quality factor of the resonance system. The accuracy attained in our measurements of a Q factor of ~106 was no worse than 1%, and the accuracy in determining the resonance frequency was ~ 20 Hz. As the temperature approaches Tc, the energy loss in the sample increases and the Q factor of the resonance system decreases from 1.6 в 106 to 105. In this case, as is seen from formula (1), the signal transmitted through the resonator decreases, thereby making it impossible to measure the dependence P( fsw) at the same position of the loops in the sample's normal state. Therefore, the distance between the loops and the resonator is reduced
INSTRUMENTS AND EXPERIMENTAL TECHNIQUES

where is the sample's geometric factor and Q0(T) and f0(T) and Q(T) and f (T) are the temperature dependences of the Q factor and resonance frequency of the empty resonator and the resonator with a sample, respectively. For a typical YBCO sample at T < 40 K, the temperature-induced change in the Q factor for the empty resonator is <10% of the change in the resonator's Q factor with a sample, and the temperatureinduced change in the resonance frequency of the empty resonator is approximately half the change in the resonance frequency of the resonator containing a sample. The value of X0 was found from the condition of the equality of the real and imaginary parts of the impedance in the normal state, R(T) = X(T), because the criterion for the normal skin effect is satisfied for all samples studied in our experiments. The sample's geometric factor depends on the sample's shape, dimensions, and arrangement in the resonator. The issue of calculating the sample's geometric factor has been considered in detail in [9, 10].
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1.2 X(T)

10

2

0.8 2 0.4

160

10

3

R(T)

140 0 10 20 T, K 30 40

0

40

80

120 T, K

Fig. 3. Temperature dependences of surface resistance R(T) and reactance X(T) in the ab planes of a YBCO single crystal at a frequency of 28.2 GHz.

Fig. 4. Low-temperature parts of curves (1) R(T) and (2) (T) = X(T)/µ0 presented in Fig. 3; (3) behavior of R(T) for another YBCO crystal from the same batch of samples.

EXAMPLES OF EXPERIMENTAL RECORDS OF THE DEPENDENCE Z(T) IN YBCO An example of experimental temperature dependences of surface resistance R(T) and reactance X(T) of an optimally doped YBa2Cu3O6.93 single crystal with a temperature of the superconducting transition Tc = 92 K is shown in Fig. 3. The crystal was grown with the technique of slow cooling from a solution in the melt with the use of a crucible made from barium zirconate BaZrO3 [10] and had the shape of a plate measuring 0.5 в 1.5 в 0.1 mm. Its geometric factor was = 9.5 k. The sample was mounted at the end of the sapphire rod so that the c axis was parallel to the microwave field; in this geometry, high-frequency currents flew in the ab crystal plane. The adjustable coupling loops allowed our measurements to be performed in the sample's normal state at 92 < T 120 K and showed that the criterion for the normal skin effect is satisfied; i.e., R(T) = X(T). The resistivity within this temperature range can be estimated from the formula (T) = 2R2(T)/µ0 0.6T µ cm. In the superconducting state, the dependence R(T) has a wide peak in the region T ~ Tc/2 typical of optimally doped YBCO crystals. Within the temperature range 10 < T < 40 K, the dependences R(T) and (T) = X(T)/µ0 are linear (Fig. 4) and their extrapolation to T = 0 K yields values of the surface resistance of R(0) 0.5 m and the field penetration depth into cuprate planes of (0) 140 nm. The possibility of performing precise measurements at temperatures of 0.4 T < 5 K allowed an unusual behavior of the surface resistance to be revealed: R(T) rapidly rises as the temperature decreases, while the

temperature dependence of reactance X(T) is linear to within the experimental accuracy. To verify this effect, we measured the surface impedance of another single crystal from the same batch of samples, which demonstrated a qualitatively identical behavior at low temperatures (see Fig. 4). At T < 1 K, the surface resistance tends to a constant value: R(0.4 Д) = 1.06 m for the first and R(0.4 Д) = 0.8 m for the second crystal. An increase in the surface resistance at T < 10 K was observed earlier only in low-quality YBCO crystals with high values of surface resistance R(0) [11] and in samples with Zn impurities, which lead to a significant decrease in critical temperature Tc [12]. Note that the rise of the dependence R(T) was accompanied by an increase in penetration depth (T) so that a minimum in curves (T) was observed in a region of 47 K. It is known that the low-temperature features in (T) in HTSCs characterized by an anisotropic d- symmetric order parameter, a coherence length of 0 ~ 1 nm, and 0 = (0) ~ 100 nm in the ab planes can be displayed at two temperatures: T* ~ (0/0)Tc ~ 1 K [13] and Tm ~ (0/0)0.5Tc ~ 10 K [14]. Temperature T* corresponds to a nonlocality-caused crossover from a linear to quadratic dependence (T), and Tm corresponds to the paramagnetic contribution from the surface bound states into (T) that increases, as the temperature decreases. Pronounced minima in the curves R(T) and (T) at T ~ 5 K and their rise at T < 5 K were also observed in GdBaCuO [15] and NdCeCuO crystals [16], the lowtemperature volumetric paramagnetism of which is evidently ensured by Gd and Nd magnetic ions.
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No peculiarities in the curves (T) are observed in Fig. 4 to within an error of measuring the field penetration depth into the sample, (T) 1 nm. This result agrees with the presence of too few bound states with a zero energy in the orientation of the YBCO crystal relative to the microwave magnetic field used in our experiments; these states are needed to form a minimum in the curve (T at T Tm [14]. Hence, an increase in surface resistance R(T) = 2 2 µ 0 3'(T)/2 at T < 10 K can be determined only by an abrupt increase in the real part of conductivity '(T), which is '(T) = nn(T)(T)e2/[m(1 + 22)] in the twofluid model, where nn is the fraction of normal carriers and is the quasi-particle relaxation time for impurities, which is 1011 s ( /(kB) 5 K) in YBCO crystals at T Tc [17, 18]. In d-symmetric superconductors with the maximum value of the slit 0, quasi-particles are excited thermally near the slit's zeros at 0 T > /(kB), but all of the known mechanisms of quasi-particle scattering by point or extended defects lead to a decrease in the conductivity, '(T) 0, at T 0 but not to its increase [18, 19]. At the same time, the presence of impurities in a d-type superconductor leads to a finite density of quasi-particles at T = 0 owing to the localization effects. These quasi-particles are scattered by the same impurities, thereby ensuring nonzero conductivity '(0) 0 at T < /(kB) [20]. It has not been determined how '(T) changes upon change from the thermal regime (T > /(kB)) to the disorder regime (T < /(kB). CONCLUSIONS A new setup for precise measurements of real R(T) and imaginary X(T) parts of the surface impedance of superconducting crystals at millimeter waves and temperatures of 0.4 T 120 K has been designed. The block diagram and the procedure for determining the numerical values of R(T) and X(T) from the resonator's Q factor and the resonance-frequency shift, which are obtained in direct measurements, are described. The dependences R(T) and X(T) measured for an optimally doped YBa2Cu3O6.93 single crystal demonstrate the known temperature behavior of the impedance of such crystals in the range 5 < T 120 K. A new effect--a considerable increase in R(T), the nature of which remains unclear--has been discovered at T < 5 K. ACKNOWLEDGMENTS We are grateful to V.F. Gantmakher for his help in the work with this paper, G.V. Merzlyakov and

S.V. Ryzhkov for their engineering assistance, and Yu.A. Nefedov and Yu.S. Barash for useful discussions. This study was supported in part by the Russian Foundation for Basic Research (projects no. 04-0217358 and 06-02-17098) and the scientific programs of the Russian Academy of Sciences. REFERENCES
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